4030:
2640:
4025:{\displaystyle {\begin{aligned}\left\langle \partial _{t}u,e^{ikx}\right\rangle &={\biggl \langle }\partial _{t}\sum _{l}{\hat {u}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }={\biggl \langle }\sum _{l}\partial _{t}{\hat {u}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }=2\pi \partial _{t}{\hat {u}}_{k},\\\left\langle f,e^{ikx}\right\rangle &={\biggl \langle }\sum _{l}{\hat {f}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }=2\pi {\hat {f}}_{k},{\text{ and}}\\\left\langle {\tfrac {1}{2}}u^{2}-\rho \partial _{x}u,\partial _{x}e^{ikx}\right\rangle &={\biggl \langle }{\tfrac {1}{2}}{\Bigl (}\sum _{p}{\hat {u}}_{p}e^{ipx}{\Bigr )}{\Bigl (}\sum _{q}{\hat {u}}_{q}e^{iqx}{\Bigr )}-\rho \partial _{x}\sum _{l}{\hat {u}}_{l}e^{ilx},\partial _{x}e^{ikx}{\biggr \rangle }\\&={\biggl \langle }{\tfrac {1}{2}}\sum _{p}\sum _{q}{\hat {u}}_{p}{\hat {u}}_{q}e^{i\left(p+q\right)x},ike^{ikx}{\biggr \rangle }-{\biggl \langle }\rho i\sum _{l}l{\hat {u}}_{l}e^{ilx},ike^{ikx}{\biggr \rangle }\\&=-{\tfrac {1}{2}}ik{\biggl \langle }\sum _{p}\sum _{q}{\hat {u}}_{p}{\hat {u}}_{q}e^{i\left(p+q\right)x},e^{ikx}{\biggr \rangle }-\rho k{\biggl \langle }\sum _{l}l{\hat {u}}_{l}e^{ilx},e^{ikx}{\biggr \rangle }\\&=-i\pi k\sum _{p+q=k}{\hat {u}}_{p}{\hat {u}}_{q}-2\pi \rho {}k^{2}{\hat {u}}_{k}.\end{aligned}}}
25:
5995:
2491:
1602:
4334:
4629:
1782:
2240:
1420:
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182:
Spectral methods can be computationally less expensive and easier to implement than finite element methods; they shine best when high accuracy is sought in simple domains with smooth solutions. However, because of their global nature, the matrices associated with step computation are dense and
166:
starting in 1969 including, but not limited to, Fourier series methods for periodic geometry problems, polynomial spectral methods for finite and unbounded geometry problems, pseudospectral methods for highly nonlinear problems, and spectral iteration methods for fast solution of steady-state
493:
1621:
2486:{\displaystyle \langle \partial _{t}u,e^{ikx}\rangle =\left\langle {\tfrac {1}{2}}u^{2}-\rho \partial _{x}u,\partial _{x}e^{ikx}\right\rangle +\left\langle f,e^{ikx}\right\rangle \quad \forall k\in \left\{-{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\},\forall t>0.}
2086:
115:
are closely related and built on the same ideas; the main difference between them is that spectral methods use basis functions that are generally nonzero over the whole domain, while finite element methods use basis functions that are nonzero only on small subdomains
4881:
of very high order, there is a similarity in the convergence properties. However, whereas the spectral method is based on the eigendecomposition of the particular boundary value problem, the finite element method does not use that information and works for arbitrary
1405:
681:
1957:
155:. When applying spectral methods to time-dependent PDEs, the solution is typically written as a sum of basis functions with time-dependent coefficients; substituting this in the PDE yields a system of ODEs in the coefficients which can be solved using any
1597:{\displaystyle \left\langle \partial _{t}u,v\right\rangle ={\Bigl \langle }\partial _{x}\left(-{\tfrac {1}{2}}u^{2}+\rho \partial _{x}u\right),v{\Bigr \rangle }+\left\langle f,v\right\rangle \quad \forall v\in {\mathcal {V}},\forall t>0}
4329:{\displaystyle 2\pi \partial _{t}{\hat {u}}_{k}=-i\pi k\sum _{p+q=k}{\hat {u}}_{p}{\hat {u}}_{q}-2\pi \rho {}k^{2}{\hat {u}}_{k}+2\pi {\hat {f}}_{k}\quad k\in \left\{-{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\},\forall t>0.}
4624:{\displaystyle \partial _{t}{\hat {u}}_{k}=-{\frac {ik}{2}}\sum _{p+q=k}{\hat {u}}_{p}{\hat {u}}_{q}-\rho {}k^{2}{\hat {u}}_{k}+{\hat {f}}_{k}\quad k\in \left\{-{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\},\forall t>0.}
2195:
842:
356:
1777:{\displaystyle \langle \partial _{t}u,v\rangle =\left\langle {\tfrac {1}{2}}u^{2}-\rho \partial _{x}u,\partial _{x}v\right\rangle +\left\langle f,v\right\rangle \quad \forall v\in {\mathcal {V}},\forall t>0.}
2578:
536:
1968:
938:
179:
approach . For very small problems, the spectral method is unique in that solutions may be written out symbolically, yielding a practical alternative to series solutions for differential equations.
2645:
1286:
336:
4760:
is infinitely differentiable, then the numerical algorithm using Fast
Fourier Transforms will converge faster than any polynomial in the grid size h. That is, for any n>0, there is a
1248:
531:
1797:
2232:
1278:
4722:
4677:
4791:
128:. Partially for this reason, spectral methods have excellent error properties, with the so-called "exponential convergence" being the fastest possible, when the solution is
2608:
187:). For larger problems and nonsmooth solutions, finite elements will generally work better due to sparse matrices and better modelling of discontinuities and sharp bends.
4828:
5022:
Jie Shen, Tao Tang and Li-Lian Wang (2011) "Spectral
Methods: Algorithms, Analysis and Applications" (Springer Series in Computational Mathematics, V. 41, Springer),
1203:
1052:
241:
4360:
1018:
6518:
136:
results (shock waves are not smooth). In the finite-element community, a method where the degree of the elements is very high or increases as the grid parameter
4868:
4848:
4758:
4053:
2632:
1072:
183:
computational efficiency will quickly suffer when there are many degrees of freedom (with some exceptions, for example if matrix applications can be written as
156:
5070:
2094:
851:
has a continuous second derivative. By the uniqueness theorem for
Fourier expansions, we must then equate the Fourier coefficients term by term, giving
6620:
5884:
692:
488:{\displaystyle \left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}\right)f(x,y)=g(x,y)\quad {\text{for all }}x,y}
6253:
4732:, and there are several transform-based techniques for evaluating it efficiently. See the references by Boyd and Canuto et al. for more details.
993:
To turn this into an algorithm, only finitely many frequencies are solved for. This introduces an error which can be shown to be proportional to
5720:
5547:
176:
6275:
514:, and can be physically interpreted as some sort of heat conduction problem, or a problem in potential theory, among other possibilities.
5710:
2503:
6676:
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6031:
5161:
6280:
5837:
5692:
5111:
6605:
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5668:
5087:
5061:
2081:{\displaystyle {\mathcal {V}}^{N}:=\operatorname {span} \left\{e^{ikx}:k\in -{\tfrac {1}{2}}N,\dots ,{\tfrac {1}{2}}N-1\right\}}
6498:
4986:
D. Gottlieb and S. Orzag (1977) "Numerical
Analysis of Spectral Methods : Theory and Applications", SIAM, Philadelphia, PA
4942:: evolution to complex geometries and applications to fluid dynamics, By Canuto, Hussaini, Quarteroni and Zang, Springer, 2007.
6351:
6149:
5385:
5155:
5013:
859:
4989:
J. Hesthaven, S. Gottlieb and D. Gottlieb (2007) "Spectral methods for time-dependent problems", Cambridge UP, Cambridge, UK
6346:
5455:
5312:
5167:
1400:{\displaystyle \partial _{t}u+u\partial _{x}u=\rho \partial _{xx}u+f\quad \forall x\in \left[0,2\pi \right),\forall t>0}
5560:
246:
6503:
5649:
5540:
68:
46:
108:) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible.
39:
6321:
5919:
5363:
847:
We have exchanged partial differentiation with an infinite sum, which is legitimate if we assume for instance that
5380:
676:{\displaystyle {\begin{aligned}f&=:\sum a_{j,k}e^{i(jx+ky)},\\g&=:\sum b_{j,k}e^{i(jx+ky)},\end{aligned}}}
6290:
5564:
5514:
5300:
4883:
1952:{\displaystyle {\mathcal {U}}^{N}:={\biggl \{}u:u(x,t)=\sum _{k=-N/2}^{N/2-1}{\hat {u}}_{k}(t)e^{ikx}{\biggr \}}}
6204:
6088:
5281:
5270:
5247:
5027:
1208:
6513:
6024:
5715:
5253:
2200:
6139:
5998:
5771:
5705:
5533:
5370:
5335:
990:
which will be equal to the mean of the resolution. This corresponds to choosing the integration constant.
6671:
6650:
6570:
6124:
5735:
5375:
5054:
1253:
6625:
6523:
6403:
5980:
5934:
5858:
5740:
5492:
5477:
5353:
5039:
4682:
4637:
4763:
6630:
6493:
6326:
6311:
6119:
6083:
5975:
5791:
5139:
5119:
5101:
6222:
6212:
6093:
6017:
5827:
5725:
5628:
5462:
5348:
5078:
4724:, this coupled system of ordinary differential equations may be integrated in time (using, e.g., a
33:
2583:
243:
is a known, complex-valued function of two real variables, and g is periodic in x and y (that is,
6585:
6560:
6378:
6367:
6078:
5924:
5700:
5504:
5482:
5467:
5450:
5358:
5343:
5259:
5124:
4910:
4874:
141:
4796:
6436:
6426:
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6129:
5955:
5899:
5863:
5424:
5195:
5047:
4957:
4905:
1137:
148:
50:
4980:
6181:
5662:
5472:
5318:
5234:
4895:
4878:
112:
93:
5658:
1173:
1023:
211:
6595:
6574:
6488:
6373:
6336:
5938:
5509:
5182:
4342:
1612:
1164:
996:
152:
89:
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8:
6398:
6134:
5904:
5842:
5556:
5276:
5190:
4983:, by Daniele Funaro, Lecture Notes in Physics, Volume 8, Springer-Verlag, Heidelberg 1992
4964:
1608:
167:
problems. The implementation of the spectral method is normally accomplished either with
85:
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6457:
6388:
6232:
6194:
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5440:
4975:
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4743:
4038:
2617:
1057:
168:
159:. Eigenvalue problems for ODEs are similarly converted to matrix eigenvalue problems .
96:. The idea is to write the solution of the differential equation as a sum of certain "
6635:
6610:
6295:
6217:
5909:
5129:
5023:
5009:
184:
16:
Class of methods used in numerical analysis and scientific computing to solve ODE/PDE
5001:
6640:
6341:
6189:
6144:
6068:
5914:
5832:
5801:
5781:
5766:
5761:
5756:
5445:
5435:
5324:
5292:
2190:{\displaystyle {\hat {u}}_{k}(t):={\frac {1}{2\pi }}\langle u(x,t),e^{ikx}\rangle }
973:
511:
5593:
6600:
6508:
6471:
6467:
6431:
6393:
6331:
6316:
6285:
6227:
6186:
6173:
6098:
6040:
6009:
5776:
5730:
5678:
5673:
5644:
5487:
5430:
4939:
4915:
2611:
1788:
172:
133:
129:
117:
5603:
4976:
A Spectral
Element Method for the Navier–Stokes Equations with Improved Accuracy
837:{\displaystyle \sum -a_{j,k}(j^{2}+k^{2})e^{i(jx+ky)}=\sum b_{j,k}e^{i(jx+ky)}.}
6565:
6544:
6462:
6452:
6263:
6170:
6103:
6063:
5965:
5817:
5618:
5265:
5212:
205:
101:
97:
6665:
5970:
5894:
5623:
5608:
5598:
5134:
4900:
2497:
163:
6383:
6237:
6178:
5960:
5613:
5583:
5306:
5223:
5200:
498:
where the expression on the left denotes the second partial derivatives of
6580:
6165:
5889:
5879:
5786:
5588:
5217:
5095:
4729:
4725:
5069:
686:
and substitute into the differential equation, we obtain this equation:
6073:
5822:
5654:
1132:
Since we're only interested in a finite window of frequencies (of size
132:. However, there are no known three-dimensional single-domain spectral
6058:
6044:
1411:
105:
6645:
6590:
201:
4735:
5403:
2573:{\displaystyle \langle e^{ilx},e^{ikx}\rangle =2\pi \delta _{lk}}
4999:
5242:
5000:
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007).
162:
Spectral methods were developed in a long series of papers by
5555:
5397:
5391:
5206:
956:
which is an explicit formula for the
Fourier coefficients
4728:
technique) to find a solution. The nonlinear term is a
933:{\displaystyle a_{j,k}=-{\frac {b_{j,k}}{j^{2}+k^{2}}}}
200:
Here we presume an understanding of basic multivariate
5008:(3rd ed.). New York: Cambridge University Press.
4581:
4557:
4286:
4262:
3660:
3417:
3173:
3084:
2443:
2419:
2291:
2053:
2029:
1659:
1482:
4856:
4836:
4799:
4766:
4746:
4685:
4640:
4371:
4345:
4064:
4041:
2643:
2620:
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2506:
2243:
2203:
2097:
1971:
1800:
1624:
1423:
1289:
1256:
1211:
1176:
1140:
algorithm. Therefore, globally the algorithm runs in
1060:
1026:
999:
862:
695:
534:
359:
249:
214:
5071:
Numerical methods for partial differential equations
1414:
coefficient. In weak conservative form this becomes
120:). Consequently, spectral methods connect variables
6621:Spectral theory of ordinary differential equations
6039:
5885:Spectral theory of ordinary differential equations
5006:Numerical Recipes: The Art of Scientific Computing
4994:Numerical Methods for the Simulation of Turbulence
4981:Polynomial Approximation of Differential Equations
4862:
4842:
4822:
4785:
4752:
4716:
4671:
4623:
4354:
4328:
4047:
4024:
2626:
2602:
2572:
2485:
2226:
2189:
2080:
1951:
1776:
1596:
1399:
1272:
1242:
1197:
1163:We wish to solve the forced, transient, nonlinear
1066:
1046:
1012:
932:
836:
675:
487:
330:
235:
6519:Schröder–Bernstein theorems for operator algebras
4969:Spectral Methods. Fundamentals in Single Domains.
3885:
3811:
3795:
3679:
3641:
3555:
3545:
3411:
3394:
3297:
3245:
3238:
3186:
3167:
3033:
2962:
2871:
2790:
2780:
2699:
1944:
1820:
1535:
1458:
6663:
331:{\displaystyle g(x,y)=g(x+2\pi ,y)=g(x,y+2\pi )}
4736:A relationship with the spectral element method
338:) then we are interested in finding a function
190:
4850:. We say that the spectral method is of order
6025:
5541:
5055:
2614:, we simplify the above three terms for each
195:
4952:A Practical Guide to Pseudospectral Methods.
4634:With Fourier transformed initial conditions
2545:
2507:
2279:
2244:
2184:
2144:
1647:
1625:
1121:by taking an inverse Fourier transform of (
6032:
6018:
5548:
5534:
5062:
5048:
4954:Cambridge University Press, Cambridge, UK
1243:{\displaystyle x\in \left[0,2\pi \right)}
69:Learn how and when to remove this message
5838:Group algebra of a locally compact group
32:This article includes a list of general
2227:{\displaystyle u\in {\mathcal {U}}^{N}}
972:With periodic boundary conditions, the
6664:
4967:, Quarteroni A., and Zang T.A. (2006)
4958:Chebyshev and Fourier Spectral Methods
2197:. This reduces the problem to finding
147:Spectral methods can be used to solve
6352:Spectral theory of normal C*-algebras
6150:Spectral theory of normal C*-algebras
6013:
5529:
5043:
4974:Javier de Frutos, Julia Novo (2000):
4830:for all sufficiently small values of
983:= 0. Therefore, we can freely choose
6347:Spectral theory of compact operators
5313:Moving particle semi-implicit method
5224:Weighted essentially non-oscillatory
4996:, Phys. Fluids Supp. II, 12, 250–257
1158:
853:
18:
4933:
1273:{\displaystyle u\in {\mathcal {U}}}
13:
6499:Cohen–Hewitt factorization theorem
5162:Finite-difference frequency-domain
4971:Springer-Verlag, Berlin Heidelberg
4780:
4609:
4373:
4314:
4072:
4035:Assemble the three terms for each
3367:
3309:
3128:
3112:
2886:
2806:
2705:
2654:
2471:
2401:
2335:
2319:
2248:
2213:
1975:
1804:
1762:
1754:
1743:
1703:
1687:
1629:
1582:
1574:
1563:
1510:
1464:
1430:
1385:
1351:
1329:
1310:
1291:
1265:
1074:is the highest frequency treated.
410:
400:
378:
368:
151:(PDEs, ODEs, eigenvalue, etc) and
84:are a class of techniques used in
38:it lacks sufficient corresponding
14:
6688:
6504:Extensions of symmetric operators
4793:such that the error is less than
4717:{\displaystyle {\hat {f}}_{k}(t)}
4672:{\displaystyle {\hat {u}}_{k}(0)}
6677:Numerical differential equations
6322:Positive operator-valued measure
5994:
5993:
5920:Topological quantum field theory
5002:"Section 20.7. Spectral Methods"
4884:elliptic boundary value problems
4786:{\displaystyle C_{n}<\infty }
1136:, say) this can be done using a
140:increases is sometimes called a
23:
6606:Rayleigh–Faber–Krahn inequality
5515:Method of fundamental solutions
5301:Smoothed-particle hydrodynamics
4541:
4246:
2400:
1742:
1562:
1350:
1097:Compute the Fourier transform (
1083:Compute the Fourier transform (
470:
5156:Alternating direction-implicit
4711:
4705:
4693:
4666:
4660:
4648:
4529:
4507:
4470:
4451:
4389:
4234:
4206:
4163:
4144:
4088:
4003:
3960:
3941:
3836:
3730:
3711:
3586:
3474:
3455:
3335:
3267:
3208:
3054:
2984:
2902:
2822:
2731:
2162:
2150:
2123:
2117:
2105:
1923:
1917:
1905:
1846:
1834:
1192:
1180:
826:
808:
773:
755:
744:
718:
661:
643:
594:
576:
467:
455:
446:
434:
325:
304:
295:
274:
265:
253:
230:
218:
1:
6514:Limiting absorption principle
5716:Uniform boundedness principle
5168:Finite-difference time-domain
4926:
1615:and using periodicity grants
976:possesses a solution only if
92:to numerically solve certain
6140:Singular value decomposition
5207:Advection upstream-splitting
2603:{\displaystyle \delta _{lk}}
1077:
510:, respectively. This is the
191:Examples of spectral methods
124:while finite elements do so
7:
6571:Hearing the shape of a drum
6254:Decomposition of a spectrum
5218:Essentially non-oscillatory
5201:Monotonic upstream-centered
5034:Spectral Methods in MATLAB.
4889:
1167:using a spectral approach.
1110:
946:
10:
6693:
6159:Special Elements/Operators
5859:Invariant subspace problem
5478:Infinite difference method
5096:Forward-time central-space
5032:Lloyd N. Trefethen (2000)
4823:{\displaystyle C_{n}h^{n}}
196:A concrete, linear example
6631:Superstrong approximation
6553:
6537:
6494:Banach algebra cohomology
6481:
6445:
6414:
6360:
6327:Projection-valued measure
6312:Borel functional calculus
6304:
6246:
6203:
6158:
6112:
6084:Projection-valued measure
6051:
5989:
5948:
5872:
5851:
5810:
5749:
5691:
5637:
5579:
5572:
5412:
5381:Poincaré–Steklov operator
5334:
5291:
5233:
5181:
5148:
5140:Method of characteristics
5110:
5086:
5077:
157:numerical method for ODEs
6223:Spectrum of a C*-algebra
6094:Spectrum of a C*-algebra
5828:Spectrum of a C*-algebra
5398:Tearing and interconnect
5392:Balancing by constraints
4992:Steven A. Orszag (1969)
4940:pp 235, Spectral Methods
6651:Wiener–Khinchin theorem
6586:Kuznetsov trace formula
6561:Almost Mathieu operator
6379:Banach function algebra
6368:Amenable Banach algebra
6125:Gelfand–Naimark theorem
6079:Noncommutative topology
5925:Noncommutative geometry
5505:Computer-assisted proof
5483:Infinite element method
5271:Gradient discretisation
4911:Spectral element method
4875:spectral element method
4362:, we finally arrive at
1205:on the periodic domain
142:spectral-element method
53:more precise citations.
6626:Sturm–Liouville theory
6524:Sherman–Takeda theorem
6404:Tomita–Takesaki theory
6179:Hermitian/Self-adjoint
6130:Gelfand representation
5981:Tomita–Takesaki theory
5956:Approximation property
5900:Calculus of variations
5493:Petrov–Galerkin method
5254:Discontinuous Galerkin
5036:SIAM, Philadelphia, PA
4950:Bengt Fornberg (1996)
4906:Pseudo-spectral method
4864:
4844:
4824:
4787:
4754:
4718:
4673:
4625:
4356:
4330:
4049:
4026:
2628:
2604:
2574:
2487:
2228:
2191:
2082:
1953:
1897:
1778:
1598:
1401:
1274:
1244:
1199:
1198:{\displaystyle u(x,0)}
1138:fast Fourier transform
1068:
1048:
1047:{\displaystyle h:=1/n}
1014:
934:
838:
677:
489:
332:
237:
236:{\displaystyle g(x,y)}
149:differential equations
113:finite-element methods
94:differential equations
6120:Gelfand–Mazur theorem
5976:Banach–Mazur distance
5939:Generalized functions
5473:Isogeometric analysis
5319:Material point method
4896:Finite element method
4879:finite element method
4865:
4845:
4825:
4788:
4755:
4740:One can show that if
4719:
4674:
4626:
4357:
4355:{\displaystyle 2\pi }
4331:
4050:
4027:
2629:
2605:
2575:
2488:
2229:
2192:
2083:
1954:
1852:
1787:To apply the Fourier–
1779:
1599:
1402:
1275:
1245:
1200:
1069:
1049:
1015:
1013:{\displaystyle h^{n}}
935:
839:
678:
490:
333:
238:
153:optimization problems
111:Spectral methods and
100:" (for example, as a
6596:Proto-value function
6575:Dirichlet eigenvalue
6489:Abstract index group
6374:Approximate identity
6337:Rigged Hilbert space
6213:Krein–Rutman theorem
6059:Involution/*-algebra
5721:Kakutani fixed-point
5706:Riesz representation
5510:Integrable algorithm
5336:Domain decomposition
4870:, for every n>0.
4854:
4834:
4797:
4764:
4744:
4683:
4638:
4369:
4343:
4339:Dividing through by
4062:
4039:
2641:
2618:
2584:
2504:
2241:
2201:
2095:
1969:
1798:
1622:
1613:Integrating by parts
1421:
1410:where ρ is the
1287:
1254:
1209:
1174:
1058:
1024:
997:
860:
693:
532:
357:
247:
212:
90:scientific computing
6399:Von Neumann algebra
6135:Polar decomposition
5905:Functional calculus
5864:Mahler's conjecture
5843:Von Neumann algebra
5557:Functional analysis
5354:Schwarz alternating
5277:Loubignac iteration
525:in Fourier series:
86:applied mathematics
6672:Numerical analysis
6529:Unbounded operator
6458:Essential spectrum
6437:Schur–Horn theorem
6427:Bauer–Fike theorem
6422:Alon–Boppana bound
6415:Finite-Dimensional
6389:Nuclear C*-algebra
6233:Spectral asymmetry
5930:Riemann hypothesis
5629:Topological vector
5500:Validated numerics
4921:Collocation method
4860:
4840:
4820:
4783:
4750:
4714:
4669:
4621:
4590:
4566:
4443:
4352:
4326:
4295:
4271:
4136:
4045:
4022:
4020:
3933:
3825:
3703:
3693:
3669:
3575:
3447:
3437:
3426:
3327:
3259:
3200:
3182:
3093:
2976:
2804:
2723:
2624:
2600:
2570:
2483:
2452:
2428:
2300:
2224:
2187:
2078:
2062:
2038:
1949:
1774:
1668:
1594:
1491:
1397:
1270:
1240:
1195:
1064:
1044:
1010:
930:
834:
673:
671:
485:
328:
233:
185:Fourier transforms
104:which is a sum of
6659:
6658:
6636:Transfer operator
6611:Spectral geometry
6296:Spectral abscissa
6276:Approximate point
6218:Normal eigenvalue
6007:
6006:
5910:Integral operator
5687:
5686:
5523:
5522:
5463:Immersed boundary
5456:Method of moments
5371:Neumann–Dirichlet
5364:abstract additive
5349:Fictitious domain
5293:Meshless/Meshfree
5177:
5176:
5079:Finite difference
5015:978-0-521-88068-8
4863:{\displaystyle n}
4843:{\displaystyle h}
4753:{\displaystyle g}
4696:
4651:
4589:
4565:
4532:
4510:
4473:
4454:
4422:
4420:
4392:
4294:
4270:
4237:
4209:
4166:
4147:
4115:
4091:
4048:{\displaystyle k}
4006:
3963:
3944:
3912:
3839:
3816:
3733:
3714:
3694:
3684:
3668:
3589:
3566:
3477:
3458:
3438:
3428:
3425:
3338:
3318:
3270:
3250:
3211:
3191:
3181:
3092:
3072:
3057:
2987:
2967:
2905:
2825:
2795:
2734:
2714:
2627:{\displaystyle k}
2451:
2427:
2299:
2142:
2108:
2061:
2037:
1908:
1667:
1607:where following
1490:
1165:Burgers' equation
1159:Nonlinear example
1108:via the formula (
1067:{\displaystyle n}
954:
953:
928:
474:
424:
392:
79:
78:
71:
6684:
6641:Transform theory
6361:Special algebras
6342:Spectral theorem
6305:Spectral Theorem
6145:Spectral theorem
6034:
6027:
6020:
6011:
6010:
5997:
5996:
5915:Jones polynomial
5833:Operator algebra
5577:
5576:
5550:
5543:
5536:
5527:
5526:
5468:Analytic element
5451:Boundary element
5344:Schur complement
5325:Particle-in-cell
5260:Spectral element
5084:
5083:
5064:
5057:
5050:
5041:
5040:
5019:
4960:by John P. Boyd.
4943:
4937:
4869:
4867:
4866:
4861:
4849:
4847:
4846:
4841:
4829:
4827:
4826:
4821:
4819:
4818:
4809:
4808:
4792:
4790:
4789:
4784:
4776:
4775:
4759:
4757:
4756:
4751:
4723:
4721:
4720:
4715:
4704:
4703:
4698:
4697:
4689:
4678:
4676:
4675:
4670:
4659:
4658:
4653:
4652:
4644:
4630:
4628:
4627:
4622:
4605:
4601:
4591:
4582:
4567:
4558:
4540:
4539:
4534:
4533:
4525:
4518:
4517:
4512:
4511:
4503:
4499:
4498:
4489:
4481:
4480:
4475:
4474:
4466:
4462:
4461:
4456:
4455:
4447:
4442:
4421:
4416:
4408:
4400:
4399:
4394:
4393:
4385:
4381:
4380:
4361:
4359:
4358:
4353:
4335:
4333:
4332:
4327:
4310:
4306:
4296:
4287:
4272:
4263:
4245:
4244:
4239:
4238:
4230:
4217:
4216:
4211:
4210:
4202:
4198:
4197:
4188:
4174:
4173:
4168:
4167:
4159:
4155:
4154:
4149:
4148:
4140:
4135:
4099:
4098:
4093:
4092:
4084:
4080:
4079:
4054:
4052:
4051:
4046:
4031:
4029:
4028:
4023:
4021:
4014:
4013:
4008:
4007:
3999:
3995:
3994:
3985:
3971:
3970:
3965:
3964:
3956:
3952:
3951:
3946:
3945:
3937:
3932:
3893:
3889:
3888:
3882:
3881:
3863:
3862:
3847:
3846:
3841:
3840:
3832:
3824:
3815:
3814:
3799:
3798:
3792:
3791:
3773:
3772:
3768:
3764:
3741:
3740:
3735:
3734:
3726:
3722:
3721:
3716:
3715:
3707:
3702:
3692:
3683:
3682:
3670:
3661:
3649:
3645:
3644:
3638:
3637:
3613:
3612:
3597:
3596:
3591:
3590:
3582:
3574:
3559:
3558:
3549:
3548:
3542:
3541:
3517:
3516:
3512:
3508:
3485:
3484:
3479:
3478:
3470:
3466:
3465:
3460:
3459:
3451:
3446:
3436:
3427:
3418:
3415:
3414:
3402:
3398:
3397:
3391:
3390:
3375:
3374:
3362:
3361:
3346:
3345:
3340:
3339:
3331:
3326:
3317:
3316:
3301:
3300:
3294:
3293:
3278:
3277:
3272:
3271:
3263:
3258:
3249:
3248:
3242:
3241:
3235:
3234:
3219:
3218:
3213:
3212:
3204:
3199:
3190:
3189:
3183:
3174:
3171:
3170:
3157:
3153:
3152:
3151:
3136:
3135:
3120:
3119:
3104:
3103:
3094:
3085:
3073:
3070:
3065:
3064:
3059:
3058:
3050:
3037:
3036:
3030:
3029:
3011:
3010:
2995:
2994:
2989:
2988:
2980:
2975:
2966:
2965:
2952:
2948:
2947:
2946:
2913:
2912:
2907:
2906:
2898:
2894:
2893:
2875:
2874:
2868:
2867:
2849:
2848:
2833:
2832:
2827:
2826:
2818:
2814:
2813:
2803:
2794:
2793:
2784:
2783:
2777:
2776:
2758:
2757:
2742:
2741:
2736:
2735:
2727:
2722:
2713:
2712:
2703:
2702:
2689:
2685:
2684:
2683:
2662:
2661:
2633:
2631:
2630:
2625:
2609:
2607:
2606:
2601:
2599:
2598:
2579:
2577:
2576:
2571:
2569:
2568:
2544:
2543:
2525:
2524:
2492:
2490:
2489:
2484:
2467:
2463:
2453:
2444:
2429:
2420:
2399:
2395:
2394:
2393:
2364:
2360:
2359:
2358:
2343:
2342:
2327:
2326:
2311:
2310:
2301:
2292:
2278:
2277:
2256:
2255:
2233:
2231:
2230:
2225:
2223:
2222:
2217:
2216:
2196:
2194:
2193:
2188:
2183:
2182:
2143:
2141:
2130:
2116:
2115:
2110:
2109:
2101:
2087:
2085:
2084:
2079:
2077:
2073:
2063:
2054:
2039:
2030:
2015:
2014:
1985:
1984:
1979:
1978:
1958:
1956:
1955:
1950:
1948:
1947:
1941:
1940:
1916:
1915:
1910:
1909:
1901:
1896:
1886:
1877:
1873:
1824:
1823:
1814:
1813:
1808:
1807:
1783:
1781:
1780:
1775:
1758:
1757:
1741:
1737:
1719:
1715:
1711:
1710:
1695:
1694:
1679:
1678:
1669:
1660:
1637:
1636:
1603:
1601:
1600:
1595:
1578:
1577:
1561:
1557:
1539:
1538:
1526:
1522:
1518:
1517:
1502:
1501:
1492:
1483:
1472:
1471:
1462:
1461:
1452:
1448:
1438:
1437:
1406:
1404:
1403:
1398:
1381:
1377:
1340:
1339:
1318:
1317:
1299:
1298:
1279:
1277:
1276:
1271:
1269:
1268:
1249:
1247:
1246:
1241:
1239:
1235:
1204:
1202:
1201:
1196:
1155:
1073:
1071:
1070:
1065:
1053:
1051:
1050:
1045:
1040:
1019:
1017:
1016:
1011:
1009:
1008:
974:Poisson equation
948:
939:
937:
936:
931:
929:
927:
926:
925:
913:
912:
902:
901:
886:
878:
877:
854:
843:
841:
840:
835:
830:
829:
799:
798:
777:
776:
743:
742:
730:
729:
717:
716:
682:
680:
679:
674:
672:
665:
664:
634:
633:
598:
597:
567:
566:
512:Poisson equation
494:
492:
491:
486:
475:
472:
430:
426:
425:
423:
422:
421:
408:
407:
398:
393:
391:
390:
389:
376:
375:
366:
337:
335:
334:
329:
242:
240:
239:
234:
82:Spectral methods
74:
67:
63:
60:
54:
49:this article by
40:inline citations
27:
26:
19:
6692:
6691:
6687:
6686:
6685:
6683:
6682:
6681:
6662:
6661:
6660:
6655:
6616:Spectral method
6601:Ramanujan graph
6549:
6533:
6509:Fredholm theory
6477:
6472:Shilov boundary
6468:Structure space
6446:Generalizations
6441:
6432:Numerical range
6410:
6394:Uniform algebra
6356:
6332:Riesz projector
6317:Min-max theorem
6300:
6286:Direct integral
6242:
6228:Spectral radius
6199:
6154:
6108:
6099:Spectral radius
6047:
6041:Spectral theory
6038:
6008:
6003:
5985:
5949:Advanced topics
5944:
5868:
5847:
5806:
5772:Hilbert–Schmidt
5745:
5736:Gelfand–Naimark
5683:
5633:
5568:
5554:
5524:
5519:
5488:Galerkin method
5431:Method of lines
5408:
5376:Neumann–Neumann
5330:
5287:
5229:
5196:High-resolution
5173:
5144:
5106:
5073:
5068:
5016:
4947:
4946:
4938:
4934:
4929:
4916:Galerkin method
4892:
4855:
4852:
4851:
4835:
4832:
4831:
4814:
4810:
4804:
4800:
4798:
4795:
4794:
4771:
4767:
4765:
4762:
4761:
4745:
4742:
4741:
4738:
4699:
4688:
4687:
4686:
4684:
4681:
4680:
4654:
4643:
4642:
4641:
4639:
4636:
4635:
4580:
4556:
4552:
4548:
4535:
4524:
4523:
4522:
4513:
4502:
4501:
4500:
4494:
4490:
4488:
4476:
4465:
4464:
4463:
4457:
4446:
4445:
4444:
4426:
4409:
4407:
4395:
4384:
4383:
4382:
4376:
4372:
4370:
4367:
4366:
4344:
4341:
4340:
4285:
4261:
4257:
4253:
4240:
4229:
4228:
4227:
4212:
4201:
4200:
4199:
4193:
4189:
4187:
4169:
4158:
4157:
4156:
4150:
4139:
4138:
4137:
4119:
4094:
4083:
4082:
4081:
4075:
4071:
4063:
4060:
4059:
4040:
4037:
4036:
4019:
4018:
4009:
3998:
3997:
3996:
3990:
3986:
3984:
3966:
3955:
3954:
3953:
3947:
3936:
3935:
3934:
3916:
3891:
3890:
3884:
3883:
3871:
3867:
3852:
3848:
3842:
3831:
3830:
3829:
3820:
3810:
3809:
3794:
3793:
3781:
3777:
3754:
3750:
3746:
3742:
3736:
3725:
3724:
3723:
3717:
3706:
3705:
3704:
3698:
3688:
3678:
3677:
3659:
3647:
3646:
3640:
3639:
3627:
3623:
3602:
3598:
3592:
3581:
3580:
3579:
3570:
3554:
3553:
3544:
3543:
3531:
3527:
3498:
3494:
3490:
3486:
3480:
3469:
3468:
3467:
3461:
3450:
3449:
3448:
3442:
3432:
3416:
3410:
3409:
3400:
3399:
3393:
3392:
3380:
3376:
3370:
3366:
3351:
3347:
3341:
3330:
3329:
3328:
3322:
3312:
3308:
3296:
3295:
3283:
3279:
3273:
3262:
3261:
3260:
3254:
3244:
3243:
3237:
3236:
3224:
3220:
3214:
3203:
3202:
3201:
3195:
3185:
3184:
3172:
3166:
3165:
3158:
3141:
3137:
3131:
3127:
3115:
3111:
3099:
3095:
3083:
3082:
3078:
3075:
3074:
3069:
3060:
3049:
3048:
3047:
3032:
3031:
3019:
3015:
3000:
2996:
2990:
2979:
2978:
2977:
2971:
2961:
2960:
2953:
2936:
2932:
2925:
2921:
2918:
2917:
2908:
2897:
2896:
2895:
2889:
2885:
2870:
2869:
2857:
2853:
2838:
2834:
2828:
2817:
2816:
2815:
2809:
2805:
2799:
2789:
2788:
2779:
2778:
2766:
2762:
2747:
2743:
2737:
2726:
2725:
2724:
2718:
2708:
2704:
2698:
2697:
2690:
2673:
2669:
2657:
2653:
2652:
2648:
2644:
2642:
2639:
2638:
2619:
2616:
2615:
2612:Kronecker delta
2591:
2587:
2585:
2582:
2581:
2561:
2557:
2533:
2529:
2514:
2510:
2505:
2502:
2501:
2442:
2418:
2414:
2410:
2383:
2379:
2372:
2368:
2348:
2344:
2338:
2334:
2322:
2318:
2306:
2302:
2290:
2289:
2285:
2267:
2263:
2251:
2247:
2242:
2239:
2238:
2218:
2212:
2211:
2210:
2202:
2199:
2198:
2172:
2168:
2134:
2129:
2111:
2100:
2099:
2098:
2096:
2093:
2092:
2052:
2028:
2004:
2000:
1999:
1995:
1980:
1974:
1973:
1972:
1970:
1967:
1966:
1943:
1942:
1930:
1926:
1911:
1900:
1899:
1898:
1882:
1878:
1869:
1856:
1819:
1818:
1809:
1803:
1802:
1801:
1799:
1796:
1795:
1789:Galerkin method
1753:
1752:
1727:
1723:
1706:
1702:
1690:
1686:
1674:
1670:
1658:
1657:
1653:
1632:
1628:
1623:
1620:
1619:
1573:
1572:
1547:
1543:
1534:
1533:
1513:
1509:
1497:
1493:
1481:
1477:
1473:
1467:
1463:
1457:
1456:
1433:
1429:
1428:
1424:
1422:
1419:
1418:
1364:
1360:
1332:
1328:
1313:
1309:
1294:
1290:
1288:
1285:
1284:
1264:
1263:
1255:
1252:
1251:
1222:
1218:
1210:
1207:
1206:
1175:
1172:
1171:
1161:
1141:
1126:
1102:
1088:
1080:
1059:
1056:
1055:
1036:
1025:
1022:
1021:
1004:
1000:
998:
995:
994:
989:
982:
968:
921:
917:
908:
904:
903:
891:
887:
885:
867:
863:
861:
858:
857:
804:
800:
788:
784:
751:
747:
738:
734:
725:
721:
706:
702:
694:
691:
690:
670:
669:
639:
635:
623:
619:
609:
603:
602:
572:
568:
556:
552:
542:
535:
533:
530:
529:
471:
417:
413:
409:
403:
399:
397:
385:
381:
377:
371:
367:
365:
364:
360:
358:
355:
354:
248:
245:
244:
213:
210:
209:
198:
193:
134:shock capturing
118:compact support
98:basis functions
75:
64:
58:
55:
45:Please help to
44:
28:
24:
17:
12:
11:
5:
6690:
6680:
6679:
6674:
6657:
6656:
6654:
6653:
6648:
6643:
6638:
6633:
6628:
6623:
6618:
6613:
6608:
6603:
6598:
6593:
6588:
6583:
6578:
6568:
6566:Corona theorem
6563:
6557:
6555:
6551:
6550:
6548:
6547:
6545:Wiener algebra
6541:
6539:
6535:
6534:
6532:
6531:
6526:
6521:
6516:
6511:
6506:
6501:
6496:
6491:
6485:
6483:
6479:
6478:
6476:
6475:
6465:
6463:Pseudospectrum
6460:
6455:
6453:Dirac spectrum
6449:
6447:
6443:
6442:
6440:
6439:
6434:
6429:
6424:
6418:
6416:
6412:
6411:
6409:
6408:
6407:
6406:
6396:
6391:
6386:
6381:
6376:
6370:
6364:
6362:
6358:
6357:
6355:
6354:
6349:
6344:
6339:
6334:
6329:
6324:
6319:
6314:
6308:
6306:
6302:
6301:
6299:
6298:
6293:
6288:
6283:
6278:
6273:
6272:
6271:
6266:
6261:
6250:
6248:
6244:
6243:
6241:
6240:
6235:
6230:
6225:
6220:
6215:
6209:
6207:
6201:
6200:
6198:
6197:
6192:
6184:
6176:
6168:
6162:
6160:
6156:
6155:
6153:
6152:
6147:
6142:
6137:
6132:
6127:
6122:
6116:
6114:
6110:
6109:
6107:
6106:
6104:Operator space
6101:
6096:
6091:
6086:
6081:
6076:
6071:
6066:
6064:Banach algebra
6061:
6055:
6053:
6052:Basic concepts
6049:
6048:
6037:
6036:
6029:
6022:
6014:
6005:
6004:
6002:
6001:
5990:
5987:
5986:
5984:
5983:
5978:
5973:
5968:
5966:Choquet theory
5963:
5958:
5952:
5950:
5946:
5945:
5943:
5942:
5932:
5927:
5922:
5917:
5912:
5907:
5902:
5897:
5892:
5887:
5882:
5876:
5874:
5870:
5869:
5867:
5866:
5861:
5855:
5853:
5849:
5848:
5846:
5845:
5840:
5835:
5830:
5825:
5820:
5818:Banach algebra
5814:
5812:
5808:
5807:
5805:
5804:
5799:
5794:
5789:
5784:
5779:
5774:
5769:
5764:
5759:
5753:
5751:
5747:
5746:
5744:
5743:
5741:Banach–Alaoglu
5738:
5733:
5728:
5723:
5718:
5713:
5708:
5703:
5697:
5695:
5689:
5688:
5685:
5684:
5682:
5681:
5676:
5671:
5669:Locally convex
5666:
5652:
5647:
5641:
5639:
5635:
5634:
5632:
5631:
5626:
5621:
5616:
5611:
5606:
5601:
5596:
5591:
5586:
5580:
5574:
5570:
5569:
5553:
5552:
5545:
5538:
5530:
5521:
5520:
5518:
5517:
5512:
5507:
5502:
5497:
5496:
5495:
5485:
5480:
5475:
5470:
5465:
5460:
5459:
5458:
5448:
5443:
5438:
5433:
5428:
5425:Pseudospectral
5422:
5416:
5414:
5410:
5409:
5407:
5406:
5401:
5395:
5389:
5383:
5378:
5373:
5368:
5367:
5366:
5361:
5351:
5346:
5340:
5338:
5332:
5331:
5329:
5328:
5322:
5316:
5310:
5304:
5297:
5295:
5289:
5288:
5286:
5285:
5279:
5274:
5268:
5263:
5257:
5251:
5245:
5239:
5237:
5235:Finite element
5231:
5230:
5228:
5227:
5221:
5215:
5213:Riemann solver
5210:
5204:
5198:
5193:
5187:
5185:
5179:
5178:
5175:
5174:
5172:
5171:
5165:
5159:
5152:
5150:
5146:
5145:
5143:
5142:
5137:
5132:
5127:
5122:
5120:Lax–Friedrichs
5116:
5114:
5108:
5107:
5105:
5104:
5102:Crank–Nicolson
5099:
5092:
5090:
5081:
5075:
5074:
5067:
5066:
5059:
5052:
5044:
5038:
5037:
5030:
5020:
5014:
4997:
4990:
4987:
4984:
4978:
4972:
4965:Hussaini M. Y.
4961:
4955:
4945:
4944:
4931:
4930:
4928:
4925:
4924:
4923:
4918:
4913:
4908:
4903:
4898:
4891:
4888:
4859:
4839:
4817:
4813:
4807:
4803:
4782:
4779:
4774:
4770:
4749:
4737:
4734:
4713:
4710:
4707:
4702:
4695:
4692:
4668:
4665:
4662:
4657:
4650:
4647:
4632:
4631:
4620:
4617:
4614:
4611:
4608:
4604:
4600:
4597:
4594:
4588:
4585:
4579:
4576:
4573:
4570:
4564:
4561:
4555:
4551:
4547:
4544:
4538:
4531:
4528:
4521:
4516:
4509:
4506:
4497:
4493:
4487:
4484:
4479:
4472:
4469:
4460:
4453:
4450:
4441:
4438:
4435:
4432:
4429:
4425:
4419:
4415:
4412:
4406:
4403:
4398:
4391:
4388:
4379:
4375:
4351:
4348:
4337:
4336:
4325:
4322:
4319:
4316:
4313:
4309:
4305:
4302:
4299:
4293:
4290:
4284:
4281:
4278:
4275:
4269:
4266:
4260:
4256:
4252:
4249:
4243:
4236:
4233:
4226:
4223:
4220:
4215:
4208:
4205:
4196:
4192:
4186:
4183:
4180:
4177:
4172:
4165:
4162:
4153:
4146:
4143:
4134:
4131:
4128:
4125:
4122:
4118:
4114:
4111:
4108:
4105:
4102:
4097:
4090:
4087:
4078:
4074:
4070:
4067:
4044:
4033:
4032:
4017:
4012:
4005:
4002:
3993:
3989:
3983:
3980:
3977:
3974:
3969:
3962:
3959:
3950:
3943:
3940:
3931:
3928:
3925:
3922:
3919:
3915:
3911:
3908:
3905:
3902:
3899:
3896:
3894:
3892:
3887:
3880:
3877:
3874:
3870:
3866:
3861:
3858:
3855:
3851:
3845:
3838:
3835:
3828:
3823:
3819:
3813:
3808:
3805:
3802:
3797:
3790:
3787:
3784:
3780:
3776:
3771:
3767:
3763:
3760:
3757:
3753:
3749:
3745:
3739:
3732:
3729:
3720:
3713:
3710:
3701:
3697:
3691:
3687:
3681:
3676:
3673:
3667:
3664:
3658:
3655:
3652:
3650:
3648:
3643:
3636:
3633:
3630:
3626:
3622:
3619:
3616:
3611:
3608:
3605:
3601:
3595:
3588:
3585:
3578:
3573:
3569:
3565:
3562:
3557:
3552:
3547:
3540:
3537:
3534:
3530:
3526:
3523:
3520:
3515:
3511:
3507:
3504:
3501:
3497:
3493:
3489:
3483:
3476:
3473:
3464:
3457:
3454:
3445:
3441:
3435:
3431:
3424:
3421:
3413:
3408:
3405:
3403:
3401:
3396:
3389:
3386:
3383:
3379:
3373:
3369:
3365:
3360:
3357:
3354:
3350:
3344:
3337:
3334:
3325:
3321:
3315:
3311:
3307:
3304:
3299:
3292:
3289:
3286:
3282:
3276:
3269:
3266:
3257:
3253:
3247:
3240:
3233:
3230:
3227:
3223:
3217:
3210:
3207:
3198:
3194:
3188:
3180:
3177:
3169:
3164:
3161:
3159:
3156:
3150:
3147:
3144:
3140:
3134:
3130:
3126:
3123:
3118:
3114:
3110:
3107:
3102:
3098:
3091:
3088:
3081:
3077:
3076:
3068:
3063:
3056:
3053:
3046:
3043:
3040:
3035:
3028:
3025:
3022:
3018:
3014:
3009:
3006:
3003:
2999:
2993:
2986:
2983:
2974:
2970:
2964:
2959:
2956:
2954:
2951:
2945:
2942:
2939:
2935:
2931:
2928:
2924:
2920:
2919:
2916:
2911:
2904:
2901:
2892:
2888:
2884:
2881:
2878:
2873:
2866:
2863:
2860:
2856:
2852:
2847:
2844:
2841:
2837:
2831:
2824:
2821:
2812:
2808:
2802:
2798:
2792:
2787:
2782:
2775:
2772:
2769:
2765:
2761:
2756:
2753:
2750:
2746:
2740:
2733:
2730:
2721:
2717:
2711:
2707:
2701:
2696:
2693:
2691:
2688:
2682:
2679:
2676:
2672:
2668:
2665:
2660:
2656:
2651:
2647:
2646:
2623:
2597:
2594:
2590:
2567:
2564:
2560:
2556:
2553:
2550:
2547:
2542:
2539:
2536:
2532:
2528:
2523:
2520:
2517:
2513:
2509:
2494:
2493:
2482:
2479:
2476:
2473:
2470:
2466:
2462:
2459:
2456:
2450:
2447:
2441:
2438:
2435:
2432:
2426:
2423:
2417:
2413:
2409:
2406:
2403:
2398:
2392:
2389:
2386:
2382:
2378:
2375:
2371:
2367:
2363:
2357:
2354:
2351:
2347:
2341:
2337:
2333:
2330:
2325:
2321:
2317:
2314:
2309:
2305:
2298:
2295:
2288:
2284:
2281:
2276:
2273:
2270:
2266:
2262:
2259:
2254:
2250:
2246:
2221:
2215:
2209:
2206:
2186:
2181:
2178:
2175:
2171:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2140:
2137:
2133:
2128:
2125:
2122:
2119:
2114:
2107:
2104:
2089:
2088:
2076:
2072:
2069:
2066:
2060:
2057:
2051:
2048:
2045:
2042:
2036:
2033:
2027:
2024:
2021:
2018:
2013:
2010:
2007:
2003:
1998:
1994:
1991:
1988:
1983:
1977:
1960:
1959:
1946:
1939:
1936:
1933:
1929:
1925:
1922:
1919:
1914:
1907:
1904:
1895:
1892:
1889:
1885:
1881:
1876:
1872:
1868:
1865:
1862:
1859:
1855:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1822:
1817:
1812:
1806:
1791:, choose both
1785:
1784:
1773:
1770:
1767:
1764:
1761:
1756:
1751:
1748:
1745:
1740:
1736:
1733:
1730:
1726:
1722:
1718:
1714:
1709:
1705:
1701:
1698:
1693:
1689:
1685:
1682:
1677:
1673:
1666:
1663:
1656:
1652:
1649:
1646:
1643:
1640:
1635:
1631:
1627:
1605:
1604:
1593:
1590:
1587:
1584:
1581:
1576:
1571:
1568:
1565:
1560:
1556:
1553:
1550:
1546:
1542:
1537:
1532:
1529:
1525:
1521:
1516:
1512:
1508:
1505:
1500:
1496:
1489:
1486:
1480:
1476:
1470:
1466:
1460:
1455:
1451:
1447:
1444:
1441:
1436:
1432:
1427:
1408:
1407:
1396:
1393:
1390:
1387:
1384:
1380:
1376:
1373:
1370:
1367:
1363:
1359:
1356:
1353:
1349:
1346:
1343:
1338:
1335:
1331:
1327:
1324:
1321:
1316:
1312:
1308:
1305:
1302:
1297:
1293:
1267:
1262:
1259:
1238:
1234:
1231:
1228:
1225:
1221:
1217:
1214:
1194:
1191:
1188:
1185:
1182:
1179:
1160:
1157:
1130:
1129:
1124:
1115:
1100:
1095:
1086:
1079:
1076:
1063:
1043:
1039:
1035:
1032:
1029:
1007:
1003:
987:
980:
960:
952:
951:
942:
940:
924:
920:
916:
911:
907:
900:
897:
894:
890:
884:
881:
876:
873:
870:
866:
845:
844:
833:
828:
825:
822:
819:
816:
813:
810:
807:
803:
797:
794:
791:
787:
783:
780:
775:
772:
769:
766:
763:
760:
757:
754:
750:
746:
741:
737:
733:
728:
724:
720:
715:
712:
709:
705:
701:
698:
684:
683:
668:
663:
660:
657:
654:
651:
648:
645:
642:
638:
632:
629:
626:
622:
618:
615:
612:
610:
608:
605:
604:
601:
596:
593:
590:
587:
584:
581:
578:
575:
571:
565:
562:
559:
555:
551:
548:
545:
543:
541:
538:
537:
496:
495:
484:
481:
478:
469:
466:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
429:
420:
416:
412:
406:
402:
396:
388:
384:
380:
374:
370:
363:
327:
324:
321:
318:
315:
312:
309:
306:
303:
300:
297:
294:
291:
288:
285:
282:
279:
276:
273:
270:
267:
264:
261:
258:
255:
252:
232:
229:
226:
223:
220:
217:
206:Fourier series
197:
194:
192:
189:
102:Fourier series
77:
76:
31:
29:
22:
15:
9:
6:
4:
3:
2:
6689:
6678:
6675:
6673:
6670:
6669:
6667:
6652:
6649:
6647:
6644:
6642:
6639:
6637:
6634:
6632:
6629:
6627:
6624:
6622:
6619:
6617:
6614:
6612:
6609:
6607:
6604:
6602:
6599:
6597:
6594:
6592:
6589:
6587:
6584:
6582:
6579:
6576:
6572:
6569:
6567:
6564:
6562:
6559:
6558:
6556:
6552:
6546:
6543:
6542:
6540:
6536:
6530:
6527:
6525:
6522:
6520:
6517:
6515:
6512:
6510:
6507:
6505:
6502:
6500:
6497:
6495:
6492:
6490:
6487:
6486:
6484:
6482:Miscellaneous
6480:
6473:
6469:
6466:
6464:
6461:
6459:
6456:
6454:
6451:
6450:
6448:
6444:
6438:
6435:
6433:
6430:
6428:
6425:
6423:
6420:
6419:
6417:
6413:
6405:
6402:
6401:
6400:
6397:
6395:
6392:
6390:
6387:
6385:
6382:
6380:
6377:
6375:
6371:
6369:
6366:
6365:
6363:
6359:
6353:
6350:
6348:
6345:
6343:
6340:
6338:
6335:
6333:
6330:
6328:
6325:
6323:
6320:
6318:
6315:
6313:
6310:
6309:
6307:
6303:
6297:
6294:
6292:
6289:
6287:
6284:
6282:
6279:
6277:
6274:
6270:
6267:
6265:
6262:
6260:
6257:
6256:
6255:
6252:
6251:
6249:
6247:Decomposition
6245:
6239:
6236:
6234:
6231:
6229:
6226:
6224:
6221:
6219:
6216:
6214:
6211:
6210:
6208:
6206:
6202:
6196:
6193:
6191:
6188:
6185:
6183:
6180:
6177:
6175:
6172:
6169:
6167:
6164:
6163:
6161:
6157:
6151:
6148:
6146:
6143:
6141:
6138:
6136:
6133:
6131:
6128:
6126:
6123:
6121:
6118:
6117:
6115:
6111:
6105:
6102:
6100:
6097:
6095:
6092:
6090:
6087:
6085:
6082:
6080:
6077:
6075:
6072:
6070:
6067:
6065:
6062:
6060:
6057:
6056:
6054:
6050:
6046:
6042:
6035:
6030:
6028:
6023:
6021:
6016:
6015:
6012:
6000:
5992:
5991:
5988:
5982:
5979:
5977:
5974:
5972:
5971:Weak topology
5969:
5967:
5964:
5962:
5959:
5957:
5954:
5953:
5951:
5947:
5940:
5936:
5933:
5931:
5928:
5926:
5923:
5921:
5918:
5916:
5913:
5911:
5908:
5906:
5903:
5901:
5898:
5896:
5895:Index theorem
5893:
5891:
5888:
5886:
5883:
5881:
5878:
5877:
5875:
5871:
5865:
5862:
5860:
5857:
5856:
5854:
5852:Open problems
5850:
5844:
5841:
5839:
5836:
5834:
5831:
5829:
5826:
5824:
5821:
5819:
5816:
5815:
5813:
5809:
5803:
5800:
5798:
5795:
5793:
5790:
5788:
5785:
5783:
5780:
5778:
5775:
5773:
5770:
5768:
5765:
5763:
5760:
5758:
5755:
5754:
5752:
5748:
5742:
5739:
5737:
5734:
5732:
5729:
5727:
5724:
5722:
5719:
5717:
5714:
5712:
5709:
5707:
5704:
5702:
5699:
5698:
5696:
5694:
5690:
5680:
5677:
5675:
5672:
5670:
5667:
5664:
5660:
5656:
5653:
5651:
5648:
5646:
5643:
5642:
5640:
5636:
5630:
5627:
5625:
5622:
5620:
5617:
5615:
5612:
5610:
5607:
5605:
5602:
5600:
5597:
5595:
5592:
5590:
5587:
5585:
5582:
5581:
5578:
5575:
5571:
5566:
5562:
5558:
5551:
5546:
5544:
5539:
5537:
5532:
5531:
5528:
5516:
5513:
5511:
5508:
5506:
5503:
5501:
5498:
5494:
5491:
5490:
5489:
5486:
5484:
5481:
5479:
5476:
5474:
5471:
5469:
5466:
5464:
5461:
5457:
5454:
5453:
5452:
5449:
5447:
5444:
5442:
5439:
5437:
5434:
5432:
5429:
5426:
5423:
5421:
5418:
5417:
5415:
5411:
5405:
5402:
5399:
5396:
5393:
5390:
5387:
5384:
5382:
5379:
5377:
5374:
5372:
5369:
5365:
5362:
5360:
5357:
5356:
5355:
5352:
5350:
5347:
5345:
5342:
5341:
5339:
5337:
5333:
5326:
5323:
5320:
5317:
5314:
5311:
5308:
5305:
5302:
5299:
5298:
5296:
5294:
5290:
5283:
5280:
5278:
5275:
5272:
5269:
5267:
5264:
5261:
5258:
5255:
5252:
5249:
5246:
5244:
5241:
5240:
5238:
5236:
5232:
5225:
5222:
5219:
5216:
5214:
5211:
5208:
5205:
5202:
5199:
5197:
5194:
5192:
5189:
5188:
5186:
5184:
5183:Finite volume
5180:
5169:
5166:
5163:
5160:
5157:
5154:
5153:
5151:
5147:
5141:
5138:
5136:
5133:
5131:
5128:
5126:
5123:
5121:
5118:
5117:
5115:
5113:
5109:
5103:
5100:
5097:
5094:
5093:
5091:
5089:
5085:
5082:
5080:
5076:
5072:
5065:
5060:
5058:
5053:
5051:
5046:
5045:
5042:
5035:
5031:
5029:
5025:
5021:
5017:
5011:
5007:
5003:
4998:
4995:
4991:
4988:
4985:
4982:
4979:
4977:
4973:
4970:
4966:
4962:
4959:
4956:
4953:
4949:
4948:
4941:
4936:
4932:
4922:
4919:
4917:
4914:
4912:
4909:
4907:
4904:
4902:
4901:Gaussian grid
4899:
4897:
4894:
4893:
4887:
4885:
4880:
4876:
4871:
4857:
4837:
4815:
4811:
4805:
4801:
4777:
4772:
4768:
4747:
4733:
4731:
4727:
4708:
4700:
4690:
4663:
4655:
4645:
4618:
4615:
4612:
4606:
4602:
4598:
4595:
4592:
4586:
4583:
4577:
4574:
4571:
4568:
4562:
4559:
4553:
4549:
4545:
4542:
4536:
4526:
4519:
4514:
4504:
4495:
4491:
4485:
4482:
4477:
4467:
4458:
4448:
4439:
4436:
4433:
4430:
4427:
4423:
4417:
4413:
4410:
4404:
4401:
4396:
4386:
4377:
4365:
4364:
4363:
4349:
4346:
4323:
4320:
4317:
4311:
4307:
4303:
4300:
4297:
4291:
4288:
4282:
4279:
4276:
4273:
4267:
4264:
4258:
4254:
4250:
4247:
4241:
4231:
4224:
4221:
4218:
4213:
4203:
4194:
4190:
4184:
4181:
4178:
4175:
4170:
4160:
4151:
4141:
4132:
4129:
4126:
4123:
4120:
4116:
4112:
4109:
4106:
4103:
4100:
4095:
4085:
4076:
4068:
4065:
4058:
4057:
4056:
4042:
4015:
4010:
4000:
3991:
3987:
3981:
3978:
3975:
3972:
3967:
3957:
3948:
3938:
3929:
3926:
3923:
3920:
3917:
3913:
3909:
3906:
3903:
3900:
3897:
3895:
3878:
3875:
3872:
3868:
3864:
3859:
3856:
3853:
3849:
3843:
3833:
3826:
3821:
3817:
3806:
3803:
3800:
3788:
3785:
3782:
3778:
3774:
3769:
3765:
3761:
3758:
3755:
3751:
3747:
3743:
3737:
3727:
3718:
3708:
3699:
3695:
3689:
3685:
3674:
3671:
3665:
3662:
3656:
3653:
3651:
3634:
3631:
3628:
3624:
3620:
3617:
3614:
3609:
3606:
3603:
3599:
3593:
3583:
3576:
3571:
3567:
3563:
3560:
3550:
3538:
3535:
3532:
3528:
3524:
3521:
3518:
3513:
3509:
3505:
3502:
3499:
3495:
3491:
3487:
3481:
3471:
3462:
3452:
3443:
3439:
3433:
3429:
3422:
3419:
3406:
3404:
3387:
3384:
3381:
3377:
3371:
3363:
3358:
3355:
3352:
3348:
3342:
3332:
3323:
3319:
3313:
3305:
3302:
3290:
3287:
3284:
3280:
3274:
3264:
3255:
3251:
3231:
3228:
3225:
3221:
3215:
3205:
3196:
3192:
3178:
3175:
3162:
3160:
3154:
3148:
3145:
3142:
3138:
3132:
3124:
3121:
3116:
3108:
3105:
3100:
3096:
3089:
3086:
3079:
3066:
3061:
3051:
3044:
3041:
3038:
3026:
3023:
3020:
3016:
3012:
3007:
3004:
3001:
2997:
2991:
2981:
2972:
2968:
2957:
2955:
2949:
2943:
2940:
2937:
2933:
2929:
2926:
2922:
2914:
2909:
2899:
2890:
2882:
2879:
2876:
2864:
2861:
2858:
2854:
2850:
2845:
2842:
2839:
2835:
2829:
2819:
2810:
2800:
2796:
2785:
2773:
2770:
2767:
2763:
2759:
2754:
2751:
2748:
2744:
2738:
2728:
2719:
2715:
2709:
2694:
2692:
2686:
2680:
2677:
2674:
2670:
2666:
2663:
2658:
2649:
2637:
2636:
2635:
2621:
2613:
2595:
2592:
2588:
2565:
2562:
2558:
2554:
2551:
2548:
2540:
2537:
2534:
2530:
2526:
2521:
2518:
2515:
2511:
2499:
2498:orthogonality
2480:
2477:
2474:
2468:
2464:
2460:
2457:
2454:
2448:
2445:
2439:
2436:
2433:
2430:
2424:
2421:
2415:
2411:
2407:
2404:
2396:
2390:
2387:
2384:
2380:
2376:
2373:
2369:
2365:
2361:
2355:
2352:
2349:
2345:
2339:
2331:
2328:
2323:
2315:
2312:
2307:
2303:
2296:
2293:
2286:
2282:
2274:
2271:
2268:
2264:
2260:
2257:
2252:
2237:
2236:
2235:
2219:
2207:
2204:
2179:
2176:
2173:
2169:
2165:
2159:
2156:
2153:
2147:
2138:
2135:
2131:
2126:
2120:
2112:
2102:
2074:
2070:
2067:
2064:
2058:
2055:
2049:
2046:
2043:
2040:
2034:
2031:
2025:
2022:
2019:
2016:
2011:
2008:
2005:
2001:
1996:
1992:
1989:
1986:
1981:
1965:
1964:
1963:
1937:
1934:
1931:
1927:
1920:
1912:
1902:
1893:
1890:
1887:
1883:
1879:
1874:
1870:
1866:
1863:
1860:
1857:
1853:
1849:
1843:
1840:
1837:
1831:
1828:
1825:
1815:
1810:
1794:
1793:
1792:
1790:
1771:
1768:
1765:
1759:
1749:
1746:
1738:
1734:
1731:
1728:
1724:
1720:
1716:
1712:
1707:
1699:
1696:
1691:
1683:
1680:
1675:
1671:
1664:
1661:
1654:
1650:
1644:
1641:
1638:
1633:
1618:
1617:
1616:
1614:
1610:
1609:inner product
1591:
1588:
1585:
1579:
1569:
1566:
1558:
1554:
1551:
1548:
1544:
1540:
1530:
1527:
1523:
1519:
1514:
1506:
1503:
1498:
1494:
1487:
1484:
1478:
1474:
1468:
1453:
1449:
1445:
1442:
1439:
1434:
1425:
1417:
1416:
1415:
1413:
1394:
1391:
1388:
1382:
1378:
1374:
1371:
1368:
1365:
1361:
1357:
1354:
1347:
1344:
1341:
1336:
1333:
1325:
1322:
1319:
1314:
1306:
1303:
1300:
1295:
1283:
1282:
1281:
1260:
1257:
1236:
1232:
1229:
1226:
1223:
1219:
1215:
1212:
1189:
1186:
1183:
1177:
1168:
1166:
1156:
1153:
1149:
1145:
1139:
1135:
1127:
1120:
1116:
1113:
1112:
1107:
1103:
1096:
1093:
1089:
1082:
1081:
1075:
1061:
1041:
1037:
1033:
1030:
1027:
1005:
1001:
991:
986:
979:
975:
970:
967:
963:
959:
950:
943:
941:
922:
918:
914:
909:
905:
898:
895:
892:
888:
882:
879:
874:
871:
868:
864:
856:
855:
852:
850:
831:
823:
820:
817:
814:
811:
805:
801:
795:
792:
789:
785:
781:
778:
770:
767:
764:
761:
758:
752:
748:
739:
735:
731:
726:
722:
713:
710:
707:
703:
699:
696:
689:
688:
687:
666:
658:
655:
652:
649:
646:
640:
636:
630:
627:
624:
620:
616:
613:
611:
606:
599:
591:
588:
585:
582:
579:
573:
569:
563:
560:
557:
553:
549:
546:
544:
539:
528:
527:
526:
524:
520:
515:
513:
509:
505:
501:
482:
479:
476:
473:for all
464:
461:
458:
452:
449:
443:
440:
437:
431:
427:
418:
414:
404:
394:
386:
382:
372:
361:
353:
352:
351:
349:
345:
341:
322:
319:
316:
313:
310:
307:
301:
298:
292:
289:
286:
283:
280:
277:
271:
268:
262:
259:
256:
250:
227:
224:
221:
215:
207:
203:
188:
186:
180:
178:
174:
170:
165:
164:Steven Orszag
160:
158:
154:
150:
145:
143:
139:
135:
131:
127:
123:
119:
114:
109:
107:
103:
99:
95:
91:
87:
83:
73:
70:
62:
52:
48:
42:
41:
35:
30:
21:
20:
6615:
6554:Applications
6384:Disk algebra
6238:Spectral gap
6113:Main results
5961:Balanced set
5935:Distribution
5873:Applications
5726:Krein–Milman
5711:Closed graph
5419:
5307:Peridynamics
5125:Lax–Wendroff
5033:
5005:
4993:
4968:
4951:
4935:
4872:
4739:
4679:and forcing
4633:
4338:
4034:
2495:
2090:
1961:
1786:
1606:
1409:
1169:
1162:
1151:
1147:
1143:
1133:
1131:
1122:
1118:
1109:
1105:
1098:
1091:
1084:
992:
984:
977:
971:
965:
961:
957:
955:
944:
848:
846:
685:
522:
518:
517:If we write
516:
507:
503:
499:
497:
347:
343:
339:
199:
181:
161:
146:
137:
125:
121:
110:
81:
80:
65:
56:
37:
6581:Heat kernel
6281:Compression
6166:Isospectral
5890:Heat kernel
5880:Hardy space
5787:Trace class
5701:Hahn–Banach
5663:Topological
5441:Collocation
4963:Canuto C.,
4730:convolution
4726:Runge Kutta
169:collocation
59:August 2013
51:introducing
6666:Categories
6259:Continuous
6074:C*-algebra
6069:B*-algebra
5823:C*-algebra
5638:Properties
5130:MacCormack
5112:Hyperbolic
5028:354071040X
4927:References
4873:Because a
4055:to obtain
2496:Using the
2234:such that
1611:notation.
1280:such that
350:) so that
34:references
6045:-algebras
5797:Unbounded
5792:Transpose
5750:Operators
5679:Separable
5674:Reflexive
5659:Algebraic
5645:Barrelled
5446:Level-set
5436:Multigrid
5386:Balancing
5088:Parabolic
4781:∞
4694:^
4649:^
4610:∀
4596:−
4575:…
4554:−
4546:∈
4530:^
4508:^
4486:ρ
4483:−
4471:^
4452:^
4424:∑
4405:−
4390:^
4374:∂
4350:π
4315:∀
4301:−
4280:…
4259:−
4251:∈
4235:^
4225:π
4207:^
4185:ρ
4182:π
4176:−
4164:^
4145:^
4117:∑
4110:π
4104:−
4089:^
4073:∂
4069:π
4004:^
3982:ρ
3979:π
3973:−
3961:^
3942:^
3914:∑
3907:π
3901:−
3837:^
3818:∑
3804:ρ
3801:−
3731:^
3712:^
3696:∑
3686:∑
3657:−
3587:^
3568:∑
3561:ρ
3551:−
3475:^
3456:^
3440:∑
3430:∑
3368:∂
3336:^
3320:∑
3310:∂
3306:ρ
3303:−
3268:^
3252:∑
3209:^
3193:∑
3129:∂
3113:∂
3109:ρ
3106:−
3071: and
3055:^
3045:π
2985:^
2969:∑
2903:^
2887:∂
2883:π
2823:^
2807:∂
2797:∑
2732:^
2716:∑
2706:∂
2655:∂
2589:δ
2559:δ
2555:π
2546:⟩
2508:⟨
2500:relation
2472:∀
2458:−
2437:…
2416:−
2408:∈
2402:∀
2336:∂
2320:∂
2316:ρ
2313:−
2280:⟩
2249:∂
2245:⟨
2208:∈
2185:⟩
2145:⟨
2139:π
2106:^
2068:−
2047:…
2026:−
2023:∈
1993:
1906:^
1891:−
1864:−
1854:∑
1763:∀
1750:∈
1744:∀
1704:∂
1688:∂
1684:ρ
1681:−
1648:⟩
1630:∂
1626:⟨
1583:∀
1570:∈
1564:∀
1511:∂
1507:ρ
1479:−
1465:∂
1431:∂
1412:viscosity
1386:∀
1375:π
1358:∈
1352:∀
1330:∂
1326:ρ
1311:∂
1292:∂
1261:∈
1233:π
1216:∈
1078:Algorithm
883:−
782:∑
700:−
697:∑
617:∑
550:∑
411:∂
401:∂
379:∂
369:∂
323:π
287:π
106:sinusoids
6646:Weyl law
6591:Lax pair
6538:Examples
6372:With an
6291:Discrete
6269:Residual
6205:Spectrum
6190:operator
6182:operator
6174:operator
6089:Spectrum
5999:Category
5811:Algebras
5693:Theorems
5650:Complete
5619:Schwartz
5565:glossary
5420:Spectral
5359:additive
5282:Smoothed
5248:Extended
4890:See also
3886:⟩
3812:⟨
3796:⟩
3680:⟨
3642:⟩
3556:⟨
3546:⟩
3412:⟨
3395:⟩
3168:⟨
3155:⟩
3080:⟨
3034:⟩
2963:⟨
2950:⟩
2923:⟨
2872:⟩
2791:⟨
2781:⟩
2700:⟨
2687:⟩
2650:⟨
2397:⟩
2370:⟨
2362:⟩
2287:⟨
1739:⟩
1725:⟨
1717:⟩
1655:⟨
1559:⟩
1545:⟨
1536:⟩
1459:⟨
1450:⟩
1426:⟨
1117:Compute
1020:, where
202:calculus
173:Galerkin
122:globally
6187:Unitary
5802:Unitary
5782:Nuclear
5767:Compact
5762:Bounded
5757:Adjoint
5731:Min–max
5624:Sobolev
5609:Nuclear
5599:Hilbert
5594:Fréchet
5559: (
5404:FETI-DP
5284:(S-FEM)
5203:(MUSCL)
5191:Godunov
2634:to see
2610:is the
1250:, find
126:locally
47:improve
6171:Normal
5777:Normal
5614:Orlicz
5604:Hölder
5584:Banach
5573:Spaces
5561:topics
5413:Others
5400:(FETI)
5394:(BDDC)
5266:Mortar
5250:(XFEM)
5243:hp-FEM
5226:(WENO)
5209:(AUSM)
5170:(FDTD)
5164:(FDFD)
5149:Others
5135:Upwind
5098:(FTCS)
5026:
5012:
2580:where
2091:where
1170:Given
130:smooth
36:, but
6264:Point
5589:Besov
5427:(DVR)
5388:(BDD)
5327:(PIC)
5321:(MPM)
5315:(MPS)
5303:(SPH)
5273:(GDM)
5262:(SEM)
5220:(ENO)
5158:(ADI)
4877:is a
1142:time
1104:) of
1090:) of
208:. If
175:or a
171:or a
6195:Unit
6043:and
5937:(or
5655:Dual
5309:(PD)
5256:(DG)
5024:ISBN
5010:ISBN
4778:<
4616:>
4321:>
2478:>
1990:span
1962:and
1769:>
1589:>
1392:>
1150:log
1054:and
521:and
506:and
204:and
88:and
1125:j,k
1101:j,k
1087:j,k
988:0,0
981:0,0
502:in
177:Tau
6668::
5563:–
5004:.
4886:.
4619:0.
4324:0.
2481:0.
2127::=
1987::=
1816::=
1772:0.
1154:).
1128:).
1114:).
1031::=
969:.
614:=:
547:=:
144:.
6577:)
6573:(
6474:)
6470:(
6033:e
6026:t
6019:v
5941:)
5665:)
5661:/
5657:(
5567:)
5549:e
5542:t
5535:v
5063:e
5056:t
5049:v
5018:.
4858:n
4838:h
4816:n
4812:h
4806:n
4802:C
4773:n
4769:C
4748:g
4712:)
4709:t
4706:(
4701:k
4691:f
4667:)
4664:0
4661:(
4656:k
4646:u
4613:t
4607:,
4603:}
4599:1
4593:N
4587:2
4584:1
4578:,
4572:,
4569:N
4563:2
4560:1
4550:{
4543:k
4537:k
4527:f
4520:+
4515:k
4505:u
4496:2
4492:k
4478:q
4468:u
4459:p
4449:u
4440:k
4437:=
4434:q
4431:+
4428:p
4418:2
4414:k
4411:i
4402:=
4397:k
4387:u
4378:t
4347:2
4318:t
4312:,
4308:}
4304:1
4298:N
4292:2
4289:1
4283:,
4277:,
4274:N
4268:2
4265:1
4255:{
4248:k
4242:k
4232:f
4222:2
4219:+
4214:k
4204:u
4195:2
4191:k
4179:2
4171:q
4161:u
4152:p
4142:u
4133:k
4130:=
4127:q
4124:+
4121:p
4113:k
4107:i
4101:=
4096:k
4086:u
4077:t
4066:2
4043:k
4016:.
4011:k
4001:u
3992:2
3988:k
3976:2
3968:q
3958:u
3949:p
3939:u
3930:k
3927:=
3924:q
3921:+
3918:p
3910:k
3904:i
3898:=
3879:x
3876:k
3873:i
3869:e
3865:,
3860:x
3857:l
3854:i
3850:e
3844:l
3834:u
3827:l
3822:l
3807:k
3789:x
3786:k
3783:i
3779:e
3775:,
3770:x
3766:)
3762:q
3759:+
3756:p
3752:(
3748:i
3744:e
3738:q
3728:u
3719:p
3709:u
3700:q
3690:p
3675:k
3672:i
3666:2
3663:1
3654:=
3635:x
3632:k
3629:i
3625:e
3621:k
3618:i
3615:,
3610:x
3607:l
3604:i
3600:e
3594:l
3584:u
3577:l
3572:l
3564:i
3539:x
3536:k
3533:i
3529:e
3525:k
3522:i
3519:,
3514:x
3510:)
3506:q
3503:+
3500:p
3496:(
3492:i
3488:e
3482:q
3472:u
3463:p
3453:u
3444:q
3434:p
3423:2
3420:1
3407:=
3388:x
3385:k
3382:i
3378:e
3372:x
3364:,
3359:x
3356:l
3353:i
3349:e
3343:l
3333:u
3324:l
3314:x
3298:)
3291:x
3288:q
3285:i
3281:e
3275:q
3265:u
3256:q
3246:(
3239:)
3232:x
3229:p
3226:i
3222:e
3216:p
3206:u
3197:p
3187:(
3179:2
3176:1
3163:=
3149:x
3146:k
3143:i
3139:e
3133:x
3125:,
3122:u
3117:x
3101:2
3097:u
3090:2
3087:1
3067:,
3062:k
3052:f
3042:2
3039:=
3027:x
3024:k
3021:i
3017:e
3013:,
3008:x
3005:l
3002:i
2998:e
2992:l
2982:f
2973:l
2958:=
2944:x
2941:k
2938:i
2934:e
2930:,
2927:f
2915:,
2910:k
2900:u
2891:t
2880:2
2877:=
2865:x
2862:k
2859:i
2855:e
2851:,
2846:x
2843:l
2840:i
2836:e
2830:l
2820:u
2811:t
2801:l
2786:=
2774:x
2771:k
2768:i
2764:e
2760:,
2755:x
2752:l
2749:i
2745:e
2739:l
2729:u
2720:l
2710:t
2695:=
2681:x
2678:k
2675:i
2671:e
2667:,
2664:u
2659:t
2622:k
2596:k
2593:l
2566:k
2563:l
2552:2
2549:=
2541:x
2538:k
2535:i
2531:e
2527:,
2522:x
2519:l
2516:i
2512:e
2475:t
2469:,
2465:}
2461:1
2455:N
2449:2
2446:1
2440:,
2434:,
2431:N
2425:2
2422:1
2412:{
2405:k
2391:x
2388:k
2385:i
2381:e
2377:,
2374:f
2366:+
2356:x
2353:k
2350:i
2346:e
2340:x
2332:,
2329:u
2324:x
2308:2
2304:u
2297:2
2294:1
2283:=
2275:x
2272:k
2269:i
2265:e
2261:,
2258:u
2253:t
2220:N
2214:U
2205:u
2180:x
2177:k
2174:i
2170:e
2166:,
2163:)
2160:t
2157:,
2154:x
2151:(
2148:u
2136:2
2132:1
2124:)
2121:t
2118:(
2113:k
2103:u
2075:}
2071:1
2065:N
2059:2
2056:1
2050:,
2044:,
2041:N
2035:2
2032:1
2020:k
2017::
2012:x
2009:k
2006:i
2002:e
1997:{
1982:N
1976:V
1945:}
1938:x
1935:k
1932:i
1928:e
1924:)
1921:t
1918:(
1913:k
1903:u
1894:1
1888:2
1884:/
1880:N
1875:2
1871:/
1867:N
1861:=
1858:k
1850:=
1847:)
1844:t
1841:,
1838:x
1835:(
1832:u
1829::
1826:u
1821:{
1811:N
1805:U
1766:t
1760:,
1755:V
1747:v
1735:v
1732:,
1729:f
1721:+
1713:v
1708:x
1700:,
1697:u
1692:x
1676:2
1672:u
1665:2
1662:1
1651:=
1645:v
1642:,
1639:u
1634:t
1592:0
1586:t
1580:,
1575:V
1567:v
1555:v
1552:,
1549:f
1541:+
1531:v
1528:,
1524:)
1520:u
1515:x
1504:+
1499:2
1495:u
1488:2
1485:1
1475:(
1469:x
1454:=
1446:v
1443:,
1440:u
1435:t
1395:0
1389:t
1383:,
1379:)
1372:2
1369:,
1366:0
1362:[
1355:x
1348:f
1345:+
1342:u
1337:x
1334:x
1323:=
1320:u
1315:x
1307:u
1304:+
1301:u
1296:t
1266:U
1258:u
1237:)
1230:2
1227:,
1224:0
1220:[
1213:x
1193:)
1190:0
1187:,
1184:x
1181:(
1178:u
1152:n
1148:n
1146:(
1144:O
1134:n
1123:a
1119:f
1111:*
1106:f
1099:a
1094:.
1092:g
1085:b
1062:n
1042:n
1038:/
1034:1
1028:h
1006:n
1002:h
985:a
978:b
966:k
964:,
962:j
958:a
949:)
947:*
945:(
923:2
919:k
915:+
910:2
906:j
899:k
896:,
893:j
889:b
880:=
875:k
872:,
869:j
865:a
849:f
832:.
827:)
824:y
821:k
818:+
815:x
812:j
809:(
806:i
802:e
796:k
793:,
790:j
786:b
779:=
774:)
771:y
768:k
765:+
762:x
759:j
756:(
753:i
749:e
745:)
740:2
736:k
732:+
727:2
723:j
719:(
714:k
711:,
708:j
704:a
667:,
662:)
659:y
656:k
653:+
650:x
647:j
644:(
641:i
637:e
631:k
628:,
625:j
621:b
607:g
600:,
595:)
592:y
589:k
586:+
583:x
580:j
577:(
574:i
570:e
564:k
561:,
558:j
554:a
540:f
523:g
519:f
508:y
504:x
500:f
483:y
480:,
477:x
468:)
465:y
462:,
459:x
456:(
453:g
450:=
447:)
444:y
441:,
438:x
435:(
432:f
428:)
419:2
415:y
405:2
395:+
387:2
383:x
373:2
362:(
348:y
346:,
344:x
342:(
340:f
326:)
320:2
317:+
314:y
311:,
308:x
305:(
302:g
299:=
296:)
293:y
290:,
284:2
281:+
278:x
275:(
272:g
269:=
266:)
263:y
260:,
257:x
254:(
251:g
231:)
228:y
225:,
222:x
219:(
216:g
138:h
116:(
72:)
66:(
61:)
57:(
43:.
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