3713:
1907:
1601:
4105:
4980:
3703:
Historically, the ADI method was developed to solve the 2D diffusion equation on a square domain using finite differences. Unlike ADI for matrix equations, ADI for parabolic equations does not require the selection of shift parameters, since the shift appearing in each iteration is determined by
5277:
Many classical implicit methods by
Peaceman-Rachford, Douglas-Gunn, D'Yakonov, Beam-Warming, Crank-Nicolson, etc., may be simplified to fundamental implicit schemes with operator-free right-hand sides. In their fundamental forms, the FADI method of second-order temporal accuracy can be related
5268:
It is possible to simplify the conventional ADI method into
Fundamental ADI method, which only has the similar operators at the left-hand sides while being operator-free at the right-hand sides. This may be regarded as the fundamental (basic) scheme of ADI method, with no more operator (to be
4758:
3891:
3217:
are non-normal matrices, it may not be possible to find near-optimal shift parameters. In this setting, a variety of strategies for generating good shift parameters can be used. These include strategies based on asymptotic results in potential theory, using the Ritz values of the matrices
5269:
reduced) at the right-hand sides, unlike most traditional implicit methods that usually consist of operators at both sides of equations. The FADI method leads to simpler, more concise and efficient update equations without degrading the accuracy of conventional ADI method.
3318:
to formulate a greedy approach, and cyclic methods, where the same small collection of shift parameters are reused until a convergence tolerance is met. When the same shift parameter is used at every iteration, ADI is equivalent to an algorithm called Smith's method.
3724:. This method results in a very complicated set of equations in multiple dimensions, which are costly to solve. The advantage of the ADI method is that the equations that have to be solved in each step have a simpler structure and can be solved efficiently with the
2484:
1649:
503:
1371:
312:
4996:
It can be shown that this method is unconditionally stable and second order in time and space. There are more refined ADI methods such as the methods of
Douglas, or the f-factor method which can be used for three or more dimensions.
2975:
3902:
3515:
4766:
6415:
4556:
3737:
4307:
3704:
parameters such as the timestep, diffusion coefficient, and grid spacing. The connection to ADI on matrix equations can be observed when one considers the action of the ADI iteration on the system at steady state.
6297:
1293:
The ADI method can still be applied when the above assumptions are not met. The use of suboptimal shift parameters may adversely affect convergence, and convergence is also affected by the non-normality of
5282:. For two- and three-dimensional heat conduction and diffusion equations, both FADI and FLOD methods may be implemented in simpler, more efficient and stable manner compared to their conventional methods.
5253:
5194:
1338:
methods, such as the
Rational Krylov Subspace Method, are observed to typically converge more rapidly than ADI in this setting, and this has led to the development of hybrid ADI-projection methods.
2016:
708:
2137:
1024:
798:
757:
2300:
2242:
565:
889:
5074:
2307:
1902:{\displaystyle {\frac {\left\|X-X^{(K)}\right\|_{2}}{\|X\|_{2}}}\leq \|r_{K}(A)\|_{2}\|r_{K}(B)^{-1}\|_{2},\quad r_{K}(M)=\prod _{j=1}^{K}{\frac {(M-\alpha _{j}I)}{(M-\beta _{j}I)}}.}
2831:
2798:
2582:
2537:
5278:
closely to the fundamental locally one-dimensional (FLOD) method, which can be upgraded to second-order temporal accuracy, such as for three-dimensional
Maxwell's equations in
4143:
3433:
6244:
4178:
3633:
1942:
4355:
2694:
1224:
167:
3175:
3146:
1135:
1106:
1053:
6365:
1642:
355:
4518:
1596:{\displaystyle X-X^{(K)}=\prod _{j=1}^{K}{\frac {(A-\alpha _{j}I)}{(A-\beta _{j}I)}}\left(X-X^{(0)}\right)\prod _{j=1}^{K}{\frac {(B-\beta _{j}I)}{(B-\alpha _{j}I)}}.}
360:
974:
836:
618:
116:
3597:
3316:
3266:
5131:
5104:
4488:
4387:
3547:
3069:
3009:
2046:
172:
4413:
4100:{\displaystyle {u_{ij}^{n+1}-u_{ij}^{n} \over \Delta t}={1 \over 2(\Delta x)^{2}}\left(\delta _{x}^{2}+\delta _{y}^{2}\right)\left(u_{ij}^{n+1}+u_{ij}^{n}\right)}
4456:
4433:
3693:
3673:
3653:
3567:
3385:
3365:
3345:
3286:
3236:
3215:
3195:
3109:
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2765:
2745:
2714:
2177:
2157:
1364:
1332:
1312:
1288:
1268:
1244:
1175:
1155:
1077:
933:
913:
638:
585:
2646:
2614:
17:
6577:
6505:
Heh, D. Y.; Tan, E. L.; Tay, W. C. (2016). "Fast
Alternating Direction Implicit Method for Efficient Transient Thermal Simulation of Integrated Circuits".
4975:{\displaystyle {u_{ij}^{n+1}-u_{ij}^{n+1/2} \over \Delta t/2}={\left(\delta _{x}^{2}u_{ij}^{n+1/2}+\delta _{y}^{2}u_{ij}^{n+1}\right) \over \Delta y^{2}}}
3438:
4753:{\displaystyle {u_{ij}^{n+1/2}-u_{ij}^{n} \over \Delta t/2}={\left(\delta _{x}^{2}u_{ij}^{n+1/2}+\delta _{y}^{2}u_{ij}^{n}\right) \over \Delta x^{2}}}
3886:{\displaystyle {\partial u \over \partial t}=\left({\partial ^{2}u \over \partial x^{2}}+{\partial ^{2}u \over \partial y^{2}}\right)=(u_{xx}+u_{yy})}
2833:, are known, the optimal shift parameter selection problem is approximately solved by finding an extremal rational function that attains the value
5823:
4190:
1346:
The problem of finding good shift parameters is nontrivial. This problem can be understood by examining the ADI error equation. After
7047:
6668:
1290:. Additionally, a priori error bounds can be computed, thereby eliminating the need to monitor the residual error in implementation.
6618:
7042:
6594:
6568:
48:
5199:
5140:
80:
The ADI method is a two step iteration process that alternately updates the column and row spaces of an approximate solution to
5508:
52:
6892:
6662:
6192:
5520:
1951:
643:
6962:
6819:
6674:
2051:
5872:
Zolotarev, D.I. (1877). "Application of elliptic functions to questions of functions deviating least and most from zero".
979:
47:, and can be formulated to construct solutions in a memory-efficient, factored form. It is also used to numerically solve
5461:
Peaceman, D. W.; Rachford Jr., H. H. (1955), "The numerical solution of parabolic and elliptic differential equations",
5799:
4536:
6464:
Tay, W. C.; Tan, E. L.; Heh, D. Y. (2014). "Fundamental
Locally One-Dimensional Method for 3-D Thermal Simulation".
6870:
762:
721:
6887:
2247:
2189:
512:
7021:
6807:
6416:"Unconditionally Stable Fundamental LOD-FDTD Method with Second-Order Temporal Accuracy and Complying Divergence"
2479:{\displaystyle \|r_{K}(A)\|_{2}\|r_{K}(B)^{-1}\|_{2}=\|r_{K}(\Lambda _{A})\|_{2}\|r_{K}(\Lambda _{B})^{-1}\|_{2}}
841:
6138:"Improved alternating-direction implicit method for solving transient three-dimensional heat diffusion problems"
6788:
6777:
6754:
5699:
5637:
Beckermann, Bernhard; Townsend, Alex (2017). "On the
Singular Values of Matrices with Displacement Structure".
5593:
Lu, An; Wachspress, E.L. (1991). "Solution of
Lyapunov equations by alternating direction implicit iteration".
5279:
5015:
4494:
2184:
6760:
892:
567:
are called shift parameters, and convergence depends strongly on the choice of these parameters. To perform
6877:
6842:
5755:
Druskin, V.; Simoncini, V. (2011). "Adaptive rational Krylov subspaces for large-scale dynamical systems".
4990:
3725:
2803:
2770:
895:. It is therefore only beneficial to use ADI when matrix-vector multiplication and linear solves involving
6208:
Lions, P. L.; Mercier, B. (December 1979). "Splitting
Algorithms for the Sum of Two Nonlinear Operators".
2542:
2497:
6882:
6561:
6298:"Fundamental Schemes for Efficient Unconditionally Stable Implicit Finite-Difference Time-Domain Methods"
6999:
6984:
6860:
6546:
4528:
4116:
3390:
6646:
6626:
6608:
4532:
4151:
3721:
3602:
3016:
3655:
is well-approximated by a low rank matrix. This is known to be true under various assumptions about
1914:
6969:
6855:
6585:
5976:
Penzl, Thilo (January 1999). "A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations".
2717:
1247:
28:
4542:
The idea behind the ADI method is to split the finite difference equations into two, one with the
7011:
6989:
6974:
6957:
6865:
6850:
6766:
6631:
4315:
2654:
1184:
498:{\displaystyle X^{(j+1)}\left(B-\alpha _{j+1}I\right)=\left(A-\alpha _{j+1}I\right)X^{(j+1/2)}-C}
125:
3151:
3122:
1111:
1082:
1029:
6931:
6702:
6554:
1608:
321:
4500:
307:{\displaystyle \left(A-\beta _{j+1}I\right)X^{(j+1/2)}=X^{(j)}\left(B-\beta _{j+1}I\right)+C.}
6979:
6825:
6741:
941:
803:
590:
83:
3572:
3291:
3241:
7016:
6689:
6469:
6430:
6322:
6217:
6149:
6079:
5985:
5846:
5492:
5422:
5109:
5082:
4461:
4360:
3716:
Stencil figure for the alternating direction implicit method in finite difference equations
3520:
3042:
2982:
2024:
1250:. Under these assumptions, near-optimal shift parameters are known for several choices of
1056:
2970:{\displaystyle Z_{K}(E,F):=\inf _{r}{\frac {\sup _{z\in E}|r(z)|}{\inf _{z\in F}|r(z)|}},}
8:
6783:
6697:
4392:
6473:
6434:
6326:
6221:
6153:
5989:
5850:
5426:
4523:
A disadvantage of the Crank–Nicolson method is that the matrix in the above equation is
4438:
7006:
6947:
6532:
6446:
6396:
6346:
6312:
6275:
6118:
5817:
5672:
5646:
5343:
5006:
4418:
3678:
3658:
3638:
3552:
3370:
3350:
3330:
3271:
3221:
3200:
3180:
3094:
3074:
3022:
2750:
2730:
2699:
2162:
2142:
1349:
1317:
1297:
1273:
1253:
1229:
1160:
1140:
1062:
918:
898:
623:
570:
64:
60:
36:
3896:
The implicit Crank–Nicolson method produces the following finite difference equation:
2619:
2587:
6636:
6507:
International Journal of Numerical Modelling: Electronic Networks, Devices and Fields
6188:
6165:
6122:
6110:
6036:
6001:
5955:
5950:
5934:"Fejér-Walsh points for rational functions and their use in the ADI iterative method"
5933:
5914:
5909:
5892:
5805:
5795:
5772:
5705:
5695:
5664:
5610:
5606:
5570:
5516:
5440:
5384:
5335:
4531:
quite costly (although efficient approximate solutions exist, for example use of the
2721:
2649:
2019:
6536:
6400:
6350:
6279:
6093:
Douglas, Jim Jr. (1962), "Alternating direction methods for three space variables",
5858:
5526:
5347:
6952:
6942:
6831:
6799:
6522:
6514:
6485:
6477:
6450:
6438:
6388:
6380:
6338:
6330:
6267:
6259:
6225:
6157:
6102:
6028:
5993:
5945:
5904:
5854:
5768:
5764:
5737:
5676:
5656:
5602:
5560:
5478:
5470:
5430:
5376:
5325:
5317:
4986:
4527:
with a band width that is generally quite large. This makes direct solution of the
3720:
The traditional method for solving the heat conduction equation numerically is the
3549:. In such a setting, it may not be feasible to store the potentially dense matrix
3510:{\displaystyle C_{1}\in \mathbb {C} ^{m\times r},C_{2}\in \mathbb {C} ^{n\times r}}
3012:
6994:
6937:
6926:
6075:
5488:
5364:
1335:
56:
5565:
5548:
5009:
scheme can be generalized. That is, we may consider general evolution equations
6772:
6719:
6481:
44:
40:
6161:
5997:
5837:
Gonchar, A.A. (1969). "Zolotarev problems connected with rational functions".
5730:
Solution of large-scale Lyapunov equations via the block modified Smith method
5483:
5435:
5410:
5380:
39:. It is a popular method for solving the large matrix equations that arise in
7036:
6641:
6442:
6384:
6263:
6169:
6114:
6040:
6005:
5959:
5918:
5809:
5776:
5709:
5668:
5614:
5574:
5444:
5388:
5339:
2180:
1178:
6334:
6813:
6730:
6707:
6185:
Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations
5507:
Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007).
5308:
Simoncini, V. (2016). "Computational Methods for Linear Matrix Equations".
5134:
2720:. In this case, the shift parameters can be expressed in closed form using
5893:"Near-circularity for the rational Zolotarev problem in the complex plane"
3569:
explicitly. A variant of ADI, called factored ADI, can be used to compute
55:
partial differential equations, and is a classic method used for modeling
6724:
6602:
4989:
and tridiagonal (banded with bandwidth 3), and is typically solved using
4524:
1945:
6576:
6527:
6392:
6342:
6271:
6245:"Efficient Algorithm for the Unconditionally Stable 3-D ADI-FDTD Method"
5330:
6490:
6106:
5660:
4302:{\displaystyle \delta _{p}^{2}u_{ij}=u_{ij+e_{p}}-2u_{ij}+u_{ij-e_{p}}}
2494:
Near-optimal shift parameters are known in certain cases, such as when
1341:
5741:
5321:
6518:
6366:"Unconditionally Stable LOD-FDTD Method for 3-D Maxwell's Equations"
6229:
6137:
6032:
5474:
6317:
5651:
1181:. These assumptions are met, for example, by the Lyapunov equation
5367:; White, Jacob (2002). "Low Rank Solution of Lyapunov Equations".
6910:
2979:
where the infimum is taken over all rational functions of degree
5506:
6749:
3712:
3011:. This approximation problem is related to several results in
6068:
Journal of the Society for Industrial and Applied Mathematics
5463:
Journal of the Society for Industrial and Applied Mathematics
6019:
Smith, R. A. (January 1968). "Matrix Equation XA + BX = C".
6904:
6898:
6713:
6054:
Douglas, J. Jr. (1955), "On the numerical integration of u
5248:{\displaystyle F_{2}={\partial ^{2} \over \partial y^{2}}}
5189:{\displaystyle F_{1}={\partial ^{2} \over \partial x^{2}}}
4497:, it can be shown that this method will be stable for any
3731:
Consider the linear diffusion equation in two dimensions,
5694:(Fourth ed.). Baltimore: Johns Hopkins University.
3635:. The effectiveness of factored ADI depends on whether
1079:. However, the ADI method performs especially well when
3114:
5515:(3rd ed.). New York: Cambridge University Press.
5509:"Section 20.3.3. Operator Splitting Methods Generally"
5409:
Benner, Peter; Li, Ren-Cang; Truhar, Ninoslav (2009).
5272:
2011:{\displaystyle \{(\alpha _{j},\beta _{j})\}_{j=1}^{K}}
1644:
results in the following bound on the relative error:
703:{\displaystyle \{(\alpha _{j},\beta _{j})\}_{j=1}^{K}}
5202:
5143:
5112:
5085:
5018:
4769:
4559:
4503:
4464:
4441:
4421:
4395:
4363:
4318:
4193:
4154:
4119:
3905:
3740:
3681:
3661:
3641:
3605:
3575:
3555:
3523:
3441:
3393:
3373:
3353:
3333:
3294:
3274:
3244:
3224:
3203:
3183:
3154:
3125:
3097:
3077:
3045:
3025:
2985:
2841:
2806:
2773:
2753:
2733:
2702:
2657:
2622:
2590:
2545:
2500:
2310:
2250:
2192:
2165:
2145:
2132:{\displaystyle \|r_{K}(A)\|_{2}\|r_{K}(B)^{-1}\|_{2}}
2054:
2027:
1954:
1917:
1652:
1611:
1374:
1352:
1320:
1300:
1276:
1256:
1232:
1187:
1163:
1143:
1114:
1085:
1065:
1032:
982:
944:
921:
901:
844:
806:
765:
724:
646:
626:
593:
573:
515:
363:
324:
175:
128:
86:
6578:
Numerical methods for partial differential equations
5460:
1342:
Shift-parameter selection and the ADI error equation
1019:{\displaystyle \sigma (A)\cap \sigma (B)=\emptyset }
118:. One ADI iteration consists of the following steps:
4546:-derivative taken implicitly and the next with the
6187:. Berlin, Heidelberg: Springer Berlin Heidelberg.
5636:
5513:Numerical Recipes: The Art of Scientific Computing
5247:
5188:
5125:
5098:
5068:
4974:
4752:
4512:
4482:
4450:
4427:
4407:
4381:
4349:
4301:
4180:is the central second difference operator for the
4172:
4137:
4099:
3885:
3687:
3667:
3647:
3627:
3591:
3561:
3541:
3509:
3427:
3379:
3359:
3339:
3310:
3280:
3260:
3230:
3209:
3189:
3169:
3140:
3103:
3083:
3063:
3031:
3003:
2969:
2825:
2792:
2759:
2739:
2708:
2688:
2640:
2608:
2576:
2531:
2478:
2294:
2236:
2171:
2151:
2131:
2040:
2010:
1936:
1901:
1636:
1595:
1358:
1326:
1306:
1282:
1262:
1238:
1218:
1169:
1149:
1129:
1100:
1071:
1047:
1018:
968:
927:
907:
883:
830:
792:
751:
702:
632:
612:
579:
559:
497:
349:
306:
161:
110:
63:in two or more dimensions. It is an example of an
6182:
3707:
7034:
6136:Chang, M. J.; Chow, L. C.; Chang, W. S. (1991),
5938:Journal of Computational and Applied Mathematics
5754:
5639:SIAM Journal on Matrix Analysis and Applications
5415:Journal of Computational and Applied Mathematics
5369:SIAM Journal on Matrix Analysis and Applications
5263:
5133:are (possibly nonlinear) operators defined on a
2924:
2884:
2871:
2696:, for example, satisfies these assumptions when
2489:
6373:IEEE Microwave and Wireless Components Letters
6252:IEEE Microwave and Wireless Components Letters
6135:
5408:
5137:space. In the diffusion example above we have
3698:
6562:
6423:IEEE Transactions on Antennas and Propagation
6305:IEEE Transactions on Antennas and Propagation
6142:Numerical Heat Transfer, Part B: Fundamentals
5689:
5595:Computers & Mathematics with Applications
5553:Computers & Mathematics with Applications
2648:are disjoint intervals on the real line. The
793:{\displaystyle B\in \mathbb {C} ^{n\times n}}
752:{\displaystyle A\in \mathbb {C} ^{m\times m}}
6207:
5592:
2467:
2427:
2418:
2388:
2376:
2343:
2334:
2311:
2295:{\displaystyle B=V_{B}\Lambda _{B}V_{B}^{*}}
2237:{\displaystyle A=V_{A}\Lambda _{A}V_{A}^{*}}
2120:
2087:
2078:
2055:
1988:
1955:
1925:
1918:
1782:
1749:
1740:
1717:
1702:
1695:
680:
647:
560:{\displaystyle (\alpha _{j+1},\beta _{j+1})}
5792:Logarithmic potentials with external fields
5411:"On the ADI method for Sylvester equations"
884:{\displaystyle {\mathcal {O}}(m^{3}+n^{3})}
70:
33:alternating-direction implicit (ADI) method
6569:
6555:
5822:: CS1 maint: location missing publisher (
5546:
2724:, and can easily be computed numerically.
6526:
6504:
6489:
6463:
6316:
6183:Hundsdorfer, Willem; Verwer, Jan (2003).
5949:
5908:
5871:
5789:
5650:
5564:
5482:
5434:
5329:
5307:
5258:
5069:{\displaystyle {\dot {u}}=F_{1}u+F_{2}u,}
3491:
3457:
3111:are disjoint disks in the complex plane.
2727:More generally, if closed, disjoint sets
774:
733:
3711:
6092:
6053:
5836:
5363:
14:
7035:
6413:
5931:
5890:
5727:
6550:
6291:
6289:
6018:
5975:
5971:
5969:
5723:
5721:
5719:
5549:"Trail to a Lyapunov equation solver"
3367:are very large, sparse matrices, and
2826:{\displaystyle \Lambda _{B}\subset F}
2793:{\displaystyle \Lambda _{A}\subset E}
976:has a unique solution if and only if
35:is an iterative method used to solve
18:Alternating direction implicit method
6820:Moving particle semi-implicit method
6731:Weighted essentially non-oscillatory
5978:SIAM Journal on Scientific Computing
5874:Zap. Imp. Akad. Nauk. St. Petersburg
5790:Saff, E.B.; Totik, V. (2013-11-11).
5632:
5630:
5628:
5626:
5624:
5588:
5586:
5584:
5456:
5454:
5404:
5402:
5400:
5398:
5359:
5357:
5303:
5301:
5299:
5297:
5295:
4985:The system of equations involved is
3115:Heuristic shift-parameter strategies
2577:{\displaystyle \Lambda _{B}\subset }
2532:{\displaystyle \Lambda _{A}\subset }
1948:. The ideal set of shift parameters
587:iterations of ADI, an initial guess
6363:
6295:
6242:
6021:SIAM Journal on Applied Mathematics
5273:Relations to other implicit methods
24:
6669:Finite-difference frequency-domain
6286:
6210:SIAM Journal on Numerical Analysis
5966:
5716:
5229:
5219:
5170:
5160:
5005:The usage of the ADI method as an
5000:
4956:
4834:
4734:
4618:
4504:
4129:
4120:
3980:
3956:
3824:
3810:
3787:
3773:
3752:
3744:
2808:
2775:
2547:
2502:
2444:
2405:
2268:
2210:
1366:iterations, the error is given by
1013:
847:
713:
25:
7059:
6466:IEICE Transactions on Electronics
5621:
5581:
5451:
5395:
5354:
5292:
4537:incomplete Cholesky factorization
4138:{\displaystyle \Delta x=\Delta y}
7048:Numerical differential equations
3428:{\displaystyle C=C_{1}C_{2}^{*}}
3071:The solution is also known when
7022:Method of fundamental solutions
6808:Smoothed-particle hydrodynamics
6498:
6457:
6414:Gan, T. H.; Tan, E. L. (2013).
6407:
6357:
6236:
6201:
6176:
6129:
6086:
6047:
6012:
5925:
5897:Journal of Approximation Theory
5884:
5865:
5859:10.1070/SM1969v007n04ABEH001107
5839:Mathematics of the USSR-Sbornik
5830:
5783:
5748:
5690:Golub, G.; Van Loan, C (1989).
4458:a shorthand for lattice points
4173:{\displaystyle \delta _{p}^{2}}
3628:{\displaystyle X\approx ZY^{*}}
3322:
1794:
7043:Partial differential equations
6663:Alternating direction-implicit
5769:10.1016/j.sysconle.2011.04.013
5683:
5547:Wachspress, Eugene L. (2008).
5540:
5499:
5280:computational electromagnetics
4550:-derivative taken implicitly,
4477:
4465:
4376:
4364:
4344:
4332:
3987:
3977:
3880:
3848:
3708:Example: 2D diffusion equation
3164:
3158:
3135:
3129:
2998:
2986:
2957:
2953:
2947:
2940:
2917:
2913:
2907:
2900:
2864:
2852:
2635:
2623:
2603:
2591:
2571:
2559:
2526:
2514:
2454:
2440:
2414:
2401:
2363:
2356:
2330:
2324:
2107:
2100:
2074:
2068:
1984:
1958:
1937:{\displaystyle \|\cdot \|_{2}}
1890:
1868:
1863:
1841:
1811:
1805:
1769:
1762:
1736:
1730:
1684:
1678:
1672:
1657:
1623:
1617:
1584:
1562:
1557:
1535:
1501:
1495:
1473:
1451:
1446:
1424:
1392:
1386:
1124:
1118:
1095:
1089:
1042:
1036:
1007:
1001:
992:
986:
878:
852:
676:
650:
605:
599:
554:
516:
484:
464:
381:
369:
342:
330:
255:
249:
236:
216:
154:
134:
13:
1:
6675:Finite-difference time-domain
5932:Starke, Gerhard (June 1993).
5891:Starke, Gerhard (July 1992).
5757:Systems & Control Letters
5285:
5264:Simplification of ADI to FADI
2490:Near-optimal shift parameters
75:
6714:Advection upstream-splitting
5951:10.1016/0377-0427(93)90291-i
5910:10.1016/0021-9045(92)90059-w
5607:10.1016/0898-1221(91)90124-m
4991:tridiagonal matrix algorithm
3726:tridiagonal matrix algorithm
2048:that minimizes the quantity
1334:(sometimes advantageously).
7:
6725:Essentially non-oscillatory
6708:Monotonic upstream-centered
5566:10.1016/j.camwa.2007.04.048
4350:{\displaystyle e_{p}=(1,0)}
3699:ADI for parabolic equations
2689:{\displaystyle AX+XA^{*}=C}
1219:{\displaystyle AX+XA^{*}=C}
162:{\displaystyle X^{(j+1/2)}}
10:
7064:
6985:Infinite difference method
6603:Forward-time central-space
6482:10.1587/transele.E97.C.636
4529:system of linear equations
3170:{\displaystyle \sigma (B)}
3141:{\displaystyle \sigma (A)}
1130:{\displaystyle \sigma (B)}
1101:{\displaystyle \sigma (A)}
1048:{\displaystyle \sigma (M)}
838:can be solved directly in
37:Sylvester matrix equations
6919:
6888:Poincaré–Steklov operator
6841:
6798:
6740:
6688:
6655:
6647:Method of characteristics
6617:
6593:
6584:
6162:10.1080/10407799108944957
5998:10.1137/s1064827598347666
5436:10.1016/j.cam.2009.08.108
5381:10.1137/s0895479801384937
4533:conjugate gradient method
3119:When less is known about
1637:{\displaystyle X^{(0)}=0}
350:{\displaystyle X^{(j+1)}}
6905:Tearing and interconnect
6899:Balancing by constraints
6443:10.1109/TAP.2013.2242036
6385:10.1109/LMWC.2006.890166
6264:10.1109/LMWC.2006.887239
4513:{\displaystyle \Delta t}
1137:are well-separated, and
935:can be applied cheaply.
620:is required, as well as
71:ADI for matrix equations
29:numerical linear algebra
7012:Computer-assisted proof
6990:Infinite element method
6778:Gradient discretisation
6468:. E-97-C (7): 636–644.
6335:10.1109/TAP.2007.913089
969:{\displaystyle AX-XB=C}
831:{\displaystyle AX-XB=C}
613:{\displaystyle X^{(0)}}
111:{\displaystyle AX-XB=C}
7000:Petrov–Galerkin method
6761:Discontinuous Galerkin
6066:by implicit methods",
5259:Fundamental ADI (FADI)
5249:
5190:
5127:
5100:
5070:
4976:
4754:
4514:
4484:
4452:
4429:
4409:
4383:
4351:
4303:
4174:
4139:
4101:
3887:
3717:
3689:
3669:
3649:
3629:
3593:
3592:{\displaystyle ZY^{*}}
3563:
3543:
3511:
3429:
3381:
3361:
3341:
3327:In many applications,
3312:
3311:{\displaystyle B^{-1}}
3282:
3262:
3261:{\displaystyle A^{-1}}
3232:
3211:
3191:
3171:
3142:
3105:
3085:
3065:
3033:
3005:
2971:
2827:
2794:
2761:
2741:
2710:
2690:
2642:
2610:
2578:
2533:
2480:
2296:
2238:
2173:
2153:
2133:
2042:
2012:
1938:
1903:
1837:
1638:
1597:
1531:
1420:
1360:
1328:
1308:
1284:
1264:
1240:
1220:
1171:
1151:
1131:
1102:
1073:
1049:
1020:
970:
929:
909:
893:Bartels-Stewart method
885:
832:
794:
753:
704:
634:
614:
581:
561:
507:
499:
351:
315:
308:
163:
112:
6980:Isogeometric analysis
6826:Material point method
6095:Numerische Mathematik
5734:PhD Diss., Rice Univ.
5250:
5191:
5128:
5126:{\displaystyle F_{2}}
5101:
5099:{\displaystyle F_{1}}
5071:
4977:
4755:
4515:
4485:
4483:{\displaystyle (i,j)}
4453:
4430:
4410:
4384:
4382:{\displaystyle (0,1)}
4352:
4304:
4175:
4140:
4102:
3888:
3722:Crank–Nicolson method
3715:
3690:
3670:
3650:
3630:
3594:
3564:
3544:
3542:{\displaystyle r=1,2}
3512:
3430:
3382:
3362:
3342:
3313:
3283:
3263:
3233:
3212:
3192:
3172:
3143:
3106:
3086:
3066:
3064:{\displaystyle F=-E.}
3034:
3006:
3004:{\displaystyle (K,K)}
2972:
2828:
2795:
2762:
2742:
2711:
2691:
2643:
2611:
2579:
2534:
2481:
2297:
2239:
2174:
2154:
2134:
2043:
2041:{\displaystyle r_{K}}
2013:
1939:
1904:
1817:
1639:
1598:
1511:
1400:
1361:
1329:
1309:
1285:
1265:
1241:
1221:
1172:
1152:
1132:
1103:
1074:
1050:
1021:
971:
930:
910:
886:
833:
795:
754:
705:
635:
615:
582:
562:
500:
352:
316:
309:
164:
120:
113:
7017:Integrable algorithm
6843:Domain decomposition
5200:
5141:
5110:
5083:
5016:
4767:
4557:
4535:preconditioned with
4501:
4462:
4439:
4419:
4393:
4361:
4316:
4191:
4152:
4117:
3903:
3738:
3679:
3659:
3639:
3603:
3573:
3553:
3521:
3439:
3391:
3371:
3351:
3331:
3292:
3272:
3242:
3222:
3201:
3181:
3152:
3123:
3095:
3075:
3043:
3023:
3015:, and was solved by
2983:
2839:
2804:
2771:
2751:
2731:
2700:
2655:
2620:
2588:
2543:
2498:
2308:
2248:
2190:
2163:
2143:
2052:
2025:
1952:
1915:
1650:
1609:
1372:
1350:
1318:
1298:
1274:
1254:
1230:
1185:
1161:
1141:
1112:
1083:
1063:
1030:
980:
942:
919:
899:
842:
804:
763:
722:
644:
624:
591:
571:
513:
361:
322:
173:
126:
84:
6861:Schwarz alternating
6784:Loubignac iteration
6474:2014IEITE..97..636T
6435:2013ITAP...61.2630G
6364:Tan, E. L. (2007).
6327:2008ITAP...56..170T
6296:Tan, E. L. (2008).
6243:Tan, E. L. (2007).
6222:1979SJNA...16..964L
6154:1991NHTB...19...69C
5990:1999SJSC...21.1401P
5851:1969SbMat...7..623G
5692:Matrix computations
5427:2009JCoAM.233.1035B
4949:
4925:
4907:
4875:
4831:
4796:
4727:
4709:
4691:
4659:
4615:
4594:
4493:After performing a
4408:{\displaystyle p=x}
4208:
4169:
4091:
4070:
4036:
4018:
3953:
3932:
3424:
3387:can be factored as
2291:
2233:
2185:eigendecompositions
2007:
699:
7007:Validated numerics
6107:10.1007/BF01386295
5728:Sabino, J (2007).
5661:10.1137/16m1096426
5484:10338.dmlcz/135399
5245:
5186:
5123:
5096:
5066:
5007:operator splitting
4972:
4926:
4911:
4876:
4861:
4800:
4773:
4750:
4710:
4695:
4660:
4645:
4598:
4563:
4510:
4495:stability analysis
4480:
4451:{\displaystyle ij}
4448:
4435:respectively (and
4425:
4405:
4379:
4347:
4299:
4194:
4170:
4155:
4135:
4097:
4074:
4047:
4022:
4004:
3936:
3909:
3883:
3718:
3685:
3665:
3645:
3625:
3589:
3559:
3539:
3507:
3425:
3410:
3377:
3357:
3337:
3308:
3278:
3258:
3228:
3207:
3187:
3167:
3138:
3101:
3081:
3061:
3029:
3001:
2967:
2938:
2898:
2879:
2823:
2790:
2757:
2737:
2722:elliptic integrals
2706:
2686:
2638:
2606:
2574:
2529:
2476:
2292:
2277:
2234:
2219:
2169:
2149:
2129:
2038:
2008:
1987:
1934:
1899:
1634:
1593:
1356:
1324:
1304:
1280:
1260:
1236:
1216:
1167:
1147:
1127:
1098:
1069:
1045:
1016:
966:
925:
905:
881:
828:
790:
749:
700:
679:
640:shift parameters,
630:
610:
577:
557:
495:
347:
304:
159:
108:
65:operator splitting
61:diffusion equation
7030:
7029:
6970:Immersed boundary
6963:Method of moments
6878:Neumann–Dirichlet
6871:abstract additive
6856:Fictitious domain
6800:Meshless/Meshfree
6684:
6683:
6586:Finite difference
6194:978-3-662-09017-6
5522:978-0-521-88068-8
5322:10.1137/130912839
5243:
5184:
5028:
4970:
4849:
4748:
4633:
4428:{\displaystyle y}
3997:
3963:
3838:
3801:
3759:
3688:{\displaystyle B}
3668:{\displaystyle A}
3648:{\displaystyle X}
3562:{\displaystyle X}
3380:{\displaystyle C}
3360:{\displaystyle B}
3340:{\displaystyle A}
3281:{\displaystyle B}
3231:{\displaystyle A}
3210:{\displaystyle B}
3190:{\displaystyle A}
3104:{\displaystyle F}
3084:{\displaystyle E}
3032:{\displaystyle E}
2962:
2923:
2883:
2870:
2760:{\displaystyle F}
2740:{\displaystyle E}
2718:positive definite
2709:{\displaystyle A}
2650:Lyapunov equation
2172:{\displaystyle B}
2152:{\displaystyle A}
2020:rational function
1894:
1712:
1588:
1477:
1359:{\displaystyle K}
1327:{\displaystyle B}
1307:{\displaystyle A}
1283:{\displaystyle B}
1263:{\displaystyle A}
1248:positive definite
1239:{\displaystyle A}
1170:{\displaystyle B}
1150:{\displaystyle A}
1072:{\displaystyle M}
928:{\displaystyle B}
908:{\displaystyle A}
633:{\displaystyle K}
580:{\displaystyle K}
16:(Redirected from
7055:
6975:Analytic element
6958:Boundary element
6851:Schur complement
6832:Particle-in-cell
6767:Spectral element
6591:
6590:
6571:
6564:
6557:
6548:
6547:
6541:
6540:
6530:
6519:10.1002/jnm.2049
6502:
6496:
6495:
6493:
6461:
6455:
6454:
6429:(5): 2630–2638.
6420:
6411:
6405:
6404:
6370:
6361:
6355:
6354:
6320:
6302:
6293:
6284:
6283:
6249:
6240:
6234:
6233:
6205:
6199:
6198:
6180:
6174:
6172:
6133:
6127:
6125:
6090:
6084:
6082:
6051:
6045:
6044:
6016:
6010:
6009:
5984:(4): 1401–1418.
5973:
5964:
5963:
5953:
5944:(1–2): 129–141.
5929:
5923:
5922:
5912:
5888:
5882:
5881:
5869:
5863:
5862:
5834:
5828:
5827:
5821:
5813:
5787:
5781:
5780:
5752:
5746:
5745:
5725:
5714:
5713:
5687:
5681:
5680:
5654:
5645:(4): 1227–1248.
5634:
5619:
5618:
5590:
5579:
5578:
5568:
5559:(8): 1653–1659.
5544:
5538:
5537:
5535:
5534:
5525:. Archived from
5503:
5497:
5495:
5486:
5458:
5449:
5448:
5438:
5421:(4): 1035–1045.
5406:
5393:
5392:
5365:Li, Jing-Rebecca
5361:
5352:
5351:
5333:
5305:
5254:
5252:
5251:
5246:
5244:
5242:
5241:
5240:
5227:
5226:
5217:
5212:
5211:
5195:
5193:
5192:
5187:
5185:
5183:
5182:
5181:
5168:
5167:
5158:
5153:
5152:
5132:
5130:
5129:
5124:
5122:
5121:
5105:
5103:
5102:
5097:
5095:
5094:
5075:
5073:
5072:
5067:
5059:
5058:
5043:
5042:
5030:
5029:
5021:
4981:
4979:
4978:
4973:
4971:
4969:
4968:
4967:
4954:
4950:
4948:
4937:
4924:
4919:
4906:
4902:
4887:
4874:
4869:
4855:
4850:
4848:
4844:
4832:
4830:
4826:
4811:
4795:
4784:
4771:
4759:
4757:
4756:
4751:
4749:
4747:
4746:
4745:
4732:
4728:
4726:
4721:
4708:
4703:
4690:
4686:
4671:
4658:
4653:
4639:
4634:
4632:
4628:
4616:
4614:
4609:
4593:
4589:
4574:
4561:
4519:
4517:
4516:
4511:
4489:
4487:
4486:
4481:
4457:
4455:
4454:
4449:
4434:
4432:
4431:
4426:
4414:
4412:
4411:
4406:
4388:
4386:
4385:
4380:
4356:
4354:
4353:
4348:
4328:
4327:
4308:
4306:
4305:
4300:
4298:
4297:
4296:
4295:
4269:
4268:
4250:
4249:
4248:
4247:
4221:
4220:
4207:
4202:
4179:
4177:
4176:
4171:
4168:
4163:
4144:
4142:
4141:
4136:
4106:
4104:
4103:
4098:
4096:
4092:
4090:
4085:
4069:
4058:
4041:
4037:
4035:
4030:
4017:
4012:
3998:
3996:
3995:
3994:
3969:
3964:
3962:
3954:
3952:
3947:
3931:
3920:
3907:
3892:
3890:
3889:
3884:
3879:
3878:
3863:
3862:
3844:
3840:
3839:
3837:
3836:
3835:
3822:
3818:
3817:
3807:
3802:
3800:
3799:
3798:
3785:
3781:
3780:
3770:
3760:
3758:
3750:
3742:
3694:
3692:
3691:
3686:
3674:
3672:
3671:
3666:
3654:
3652:
3651:
3646:
3634:
3632:
3631:
3626:
3624:
3623:
3598:
3596:
3595:
3590:
3588:
3587:
3568:
3566:
3565:
3560:
3548:
3546:
3545:
3540:
3516:
3514:
3513:
3508:
3506:
3505:
3494:
3485:
3484:
3472:
3471:
3460:
3451:
3450:
3434:
3432:
3431:
3426:
3423:
3418:
3409:
3408:
3386:
3384:
3383:
3378:
3366:
3364:
3363:
3358:
3346:
3344:
3343:
3338:
3317:
3315:
3314:
3309:
3307:
3306:
3287:
3285:
3284:
3279:
3267:
3265:
3264:
3259:
3257:
3256:
3237:
3235:
3234:
3229:
3216:
3214:
3213:
3208:
3196:
3194:
3193:
3188:
3176:
3174:
3173:
3168:
3147:
3145:
3144:
3139:
3110:
3108:
3107:
3102:
3090:
3088:
3087:
3082:
3070:
3068:
3067:
3062:
3038:
3036:
3035:
3030:
3013:potential theory
3010:
3008:
3007:
3002:
2976:
2974:
2973:
2968:
2963:
2961:
2960:
2943:
2937:
2921:
2920:
2903:
2897:
2881:
2878:
2851:
2850:
2832:
2830:
2829:
2824:
2816:
2815:
2799:
2797:
2796:
2791:
2783:
2782:
2766:
2764:
2763:
2758:
2746:
2744:
2743:
2738:
2715:
2713:
2712:
2707:
2695:
2693:
2692:
2687:
2679:
2678:
2647:
2645:
2644:
2641:{\displaystyle }
2639:
2615:
2613:
2612:
2609:{\displaystyle }
2607:
2583:
2581:
2580:
2575:
2555:
2554:
2538:
2536:
2535:
2530:
2510:
2509:
2485:
2483:
2482:
2477:
2475:
2474:
2465:
2464:
2452:
2451:
2439:
2438:
2426:
2425:
2413:
2412:
2400:
2399:
2384:
2383:
2374:
2373:
2355:
2354:
2342:
2341:
2323:
2322:
2301:
2299:
2298:
2293:
2290:
2285:
2276:
2275:
2266:
2265:
2243:
2241:
2240:
2235:
2232:
2227:
2218:
2217:
2208:
2207:
2178:
2176:
2175:
2170:
2158:
2156:
2155:
2150:
2138:
2136:
2135:
2130:
2128:
2127:
2118:
2117:
2099:
2098:
2086:
2085:
2067:
2066:
2047:
2045:
2044:
2039:
2037:
2036:
2017:
2015:
2014:
2009:
2006:
2001:
1983:
1982:
1970:
1969:
1943:
1941:
1940:
1935:
1933:
1932:
1908:
1906:
1905:
1900:
1895:
1893:
1886:
1885:
1866:
1859:
1858:
1839:
1836:
1831:
1804:
1803:
1790:
1789:
1780:
1779:
1761:
1760:
1748:
1747:
1729:
1728:
1713:
1711:
1710:
1709:
1693:
1692:
1687:
1683:
1682:
1681:
1654:
1643:
1641:
1640:
1635:
1627:
1626:
1602:
1600:
1599:
1594:
1589:
1587:
1580:
1579:
1560:
1553:
1552:
1533:
1530:
1525:
1510:
1506:
1505:
1504:
1478:
1476:
1469:
1468:
1449:
1442:
1441:
1422:
1419:
1414:
1396:
1395:
1365:
1363:
1362:
1357:
1333:
1331:
1330:
1325:
1313:
1311:
1310:
1305:
1289:
1287:
1286:
1281:
1269:
1267:
1266:
1261:
1245:
1243:
1242:
1237:
1225:
1223:
1222:
1217:
1209:
1208:
1176:
1174:
1173:
1168:
1156:
1154:
1153:
1148:
1136:
1134:
1133:
1128:
1107:
1105:
1104:
1099:
1078:
1076:
1075:
1070:
1054:
1052:
1051:
1046:
1025:
1023:
1022:
1017:
975:
973:
972:
967:
934:
932:
931:
926:
914:
912:
911:
906:
890:
888:
887:
882:
877:
876:
864:
863:
851:
850:
837:
835:
834:
829:
799:
797:
796:
791:
789:
788:
777:
758:
756:
755:
750:
748:
747:
736:
709:
707:
706:
701:
698:
693:
675:
674:
662:
661:
639:
637:
636:
631:
619:
617:
616:
611:
609:
608:
586:
584:
583:
578:
566:
564:
563:
558:
553:
552:
534:
533:
504:
502:
501:
496:
488:
487:
480:
458:
454:
450:
449:
420:
416:
412:
411:
385:
384:
356:
354:
353:
348:
346:
345:
313:
311:
310:
305:
294:
290:
286:
285:
259:
258:
240:
239:
232:
210:
206:
202:
201:
168:
166:
165:
160:
158:
157:
150:
117:
115:
114:
109:
59:and solving the
21:
7063:
7062:
7058:
7057:
7056:
7054:
7053:
7052:
7033:
7032:
7031:
7026:
6995:Galerkin method
6938:Method of lines
6915:
6883:Neumann–Neumann
6837:
6794:
6736:
6703:High-resolution
6680:
6651:
6613:
6580:
6575:
6545:
6544:
6503:
6499:
6462:
6458:
6418:
6412:
6408:
6368:
6362:
6358:
6300:
6294:
6287:
6247:
6241:
6237:
6230:10.1137/0716071
6206:
6202:
6195:
6181:
6177:
6134:
6130:
6091:
6087:
6065:
6061:
6057:
6052:
6048:
6033:10.1137/0116017
6017:
6013:
5974:
5967:
5930:
5926:
5889:
5885:
5870:
5866:
5835:
5831:
5815:
5814:
5802:
5788:
5784:
5753:
5749:
5726:
5717:
5702:
5688:
5684:
5635:
5622:
5591:
5582:
5545:
5541:
5532:
5530:
5523:
5504:
5500:
5475:10.1137/0103003
5459:
5452:
5407:
5396:
5362:
5355:
5306:
5293:
5288:
5275:
5266:
5261:
5236:
5232:
5228:
5222:
5218:
5216:
5207:
5203:
5201:
5198:
5197:
5177:
5173:
5169:
5163:
5159:
5157:
5148:
5144:
5142:
5139:
5138:
5117:
5113:
5111:
5108:
5107:
5090:
5086:
5084:
5081:
5080:
5054:
5050:
5038:
5034:
5020:
5019:
5017:
5014:
5013:
5003:
5001:Generalizations
4963:
4959:
4955:
4938:
4930:
4920:
4915:
4898:
4888:
4880:
4870:
4865:
4860:
4856:
4854:
4840:
4833:
4822:
4812:
4804:
4785:
4777:
4772:
4770:
4768:
4765:
4764:
4741:
4737:
4733:
4722:
4714:
4704:
4699:
4682:
4672:
4664:
4654:
4649:
4644:
4640:
4638:
4624:
4617:
4610:
4602:
4585:
4575:
4567:
4562:
4560:
4558:
4555:
4554:
4502:
4499:
4498:
4463:
4460:
4459:
4440:
4437:
4436:
4420:
4417:
4416:
4394:
4391:
4390:
4362:
4359:
4358:
4323:
4319:
4317:
4314:
4313:
4291:
4287:
4277:
4273:
4261:
4257:
4243:
4239:
4229:
4225:
4213:
4209:
4203:
4198:
4192:
4189:
4188:
4184:-th coordinate
4164:
4159:
4153:
4150:
4149:
4118:
4115:
4114:
4086:
4078:
4059:
4051:
4046:
4042:
4031:
4026:
4013:
4008:
4003:
3999:
3990:
3986:
3973:
3968:
3955:
3948:
3940:
3921:
3913:
3908:
3906:
3904:
3901:
3900:
3871:
3867:
3855:
3851:
3831:
3827:
3823:
3813:
3809:
3808:
3806:
3794:
3790:
3786:
3776:
3772:
3771:
3769:
3768:
3764:
3751:
3743:
3741:
3739:
3736:
3735:
3710:
3701:
3680:
3677:
3676:
3660:
3657:
3656:
3640:
3637:
3636:
3619:
3615:
3604:
3601:
3600:
3583:
3579:
3574:
3571:
3570:
3554:
3551:
3550:
3522:
3519:
3518:
3495:
3490:
3489:
3480:
3476:
3461:
3456:
3455:
3446:
3442:
3440:
3437:
3436:
3419:
3414:
3404:
3400:
3392:
3389:
3388:
3372:
3369:
3368:
3352:
3349:
3348:
3332:
3329:
3328:
3325:
3299:
3295:
3293:
3290:
3289:
3273:
3270:
3269:
3249:
3245:
3243:
3240:
3239:
3223:
3220:
3219:
3202:
3199:
3198:
3182:
3179:
3178:
3153:
3150:
3149:
3124:
3121:
3120:
3117:
3096:
3093:
3092:
3076:
3073:
3072:
3044:
3041:
3040:
3024:
3021:
3020:
2984:
2981:
2980:
2956:
2939:
2927:
2922:
2916:
2899:
2887:
2882:
2880:
2874:
2846:
2842:
2840:
2837:
2836:
2811:
2807:
2805:
2802:
2801:
2778:
2774:
2772:
2769:
2768:
2752:
2749:
2748:
2732:
2729:
2728:
2701:
2698:
2697:
2674:
2670:
2656:
2653:
2652:
2621:
2618:
2617:
2589:
2586:
2585:
2550:
2546:
2544:
2541:
2540:
2505:
2501:
2499:
2496:
2495:
2492:
2470:
2466:
2457:
2453:
2447:
2443:
2434:
2430:
2421:
2417:
2408:
2404:
2395:
2391:
2379:
2375:
2366:
2362:
2350:
2346:
2337:
2333:
2318:
2314:
2309:
2306:
2305:
2286:
2281:
2271:
2267:
2261:
2257:
2249:
2246:
2245:
2228:
2223:
2213:
2209:
2203:
2199:
2191:
2188:
2187:
2181:normal matrices
2164:
2161:
2160:
2144:
2141:
2140:
2123:
2119:
2110:
2106:
2094:
2090:
2081:
2077:
2062:
2058:
2053:
2050:
2049:
2032:
2028:
2026:
2023:
2022:
2002:
1991:
1978:
1974:
1965:
1961:
1953:
1950:
1949:
1928:
1924:
1916:
1913:
1912:
1881:
1877:
1867:
1854:
1850:
1840:
1838:
1832:
1821:
1799:
1795:
1785:
1781:
1772:
1768:
1756:
1752:
1743:
1739:
1724:
1720:
1705:
1701:
1694:
1688:
1671:
1667:
1660:
1656:
1655:
1653:
1651:
1648:
1647:
1616:
1612:
1610:
1607:
1606:
1575:
1571:
1561:
1548:
1544:
1534:
1532:
1526:
1515:
1494:
1490:
1483:
1479:
1464:
1460:
1450:
1437:
1433:
1423:
1421:
1415:
1404:
1385:
1381:
1373:
1370:
1369:
1351:
1348:
1347:
1344:
1336:Krylov subspace
1319:
1316:
1315:
1299:
1296:
1295:
1275:
1272:
1271:
1255:
1252:
1251:
1231:
1228:
1227:
1204:
1200:
1186:
1183:
1182:
1179:normal matrices
1162:
1159:
1158:
1142:
1139:
1138:
1113:
1110:
1109:
1084:
1081:
1080:
1064:
1061:
1060:
1031:
1028:
1027:
981:
978:
977:
943:
940:
939:
920:
917:
916:
900:
897:
896:
872:
868:
859:
855:
846:
845:
843:
840:
839:
805:
802:
801:
778:
773:
772:
764:
761:
760:
737:
732:
731:
723:
720:
719:
716:
714:When to use ADI
694:
683:
670:
666:
657:
653:
645:
642:
641:
625:
622:
621:
598:
594:
592:
589:
588:
572:
569:
568:
542:
538:
523:
519:
514:
511:
510:
476:
463:
459:
439:
435:
428:
424:
401:
397:
390:
386:
368:
364:
362:
359:
358:
329:
325:
323:
320:
319:
275:
271:
264:
260:
248:
244:
228:
215:
211:
191:
187:
180:
176:
174:
171:
170:
146:
133:
129:
127:
124:
123:
85:
82:
81:
78:
73:
57:heat conduction
23:
22:
15:
12:
11:
5:
7061:
7051:
7050:
7045:
7028:
7027:
7025:
7024:
7019:
7014:
7009:
7004:
7003:
7002:
6992:
6987:
6982:
6977:
6972:
6967:
6966:
6965:
6955:
6950:
6945:
6940:
6935:
6932:Pseudospectral
6929:
6923:
6921:
6917:
6916:
6914:
6913:
6908:
6902:
6896:
6890:
6885:
6880:
6875:
6874:
6873:
6868:
6858:
6853:
6847:
6845:
6839:
6838:
6836:
6835:
6829:
6823:
6817:
6811:
6804:
6802:
6796:
6795:
6793:
6792:
6786:
6781:
6775:
6770:
6764:
6758:
6752:
6746:
6744:
6742:Finite element
6738:
6737:
6735:
6734:
6728:
6722:
6720:Riemann solver
6717:
6711:
6705:
6700:
6694:
6692:
6686:
6685:
6682:
6681:
6679:
6678:
6672:
6666:
6659:
6657:
6653:
6652:
6650:
6649:
6644:
6639:
6634:
6629:
6627:Lax–Friedrichs
6623:
6621:
6615:
6614:
6612:
6611:
6609:Crank–Nicolson
6606:
6599:
6597:
6588:
6582:
6581:
6574:
6573:
6566:
6559:
6551:
6543:
6542:
6497:
6456:
6406:
6356:
6311:(1): 170–177.
6285:
6235:
6216:(6): 964–979.
6200:
6193:
6175:
6128:
6085:
6063:
6059:
6055:
6046:
6027:(1): 198–201.
6011:
5965:
5924:
5903:(1): 115–130.
5883:
5864:
5845:(4): 623–635.
5829:
5800:
5782:
5763:(8): 546–560.
5747:
5715:
5700:
5682:
5620:
5580:
5539:
5521:
5498:
5450:
5394:
5375:(1): 260–280.
5353:
5316:(3): 377–441.
5290:
5289:
5287:
5284:
5274:
5271:
5265:
5262:
5260:
5257:
5239:
5235:
5231:
5225:
5221:
5215:
5210:
5206:
5180:
5176:
5172:
5166:
5162:
5156:
5151:
5147:
5120:
5116:
5093:
5089:
5077:
5076:
5065:
5062:
5057:
5053:
5049:
5046:
5041:
5037:
5033:
5027:
5024:
5002:
4999:
4983:
4982:
4966:
4962:
4958:
4953:
4947:
4944:
4941:
4936:
4933:
4929:
4923:
4918:
4914:
4910:
4905:
4901:
4897:
4894:
4891:
4886:
4883:
4879:
4873:
4868:
4864:
4859:
4853:
4847:
4843:
4839:
4836:
4829:
4825:
4821:
4818:
4815:
4810:
4807:
4803:
4799:
4794:
4791:
4788:
4783:
4780:
4776:
4761:
4760:
4744:
4740:
4736:
4731:
4725:
4720:
4717:
4713:
4707:
4702:
4698:
4694:
4689:
4685:
4681:
4678:
4675:
4670:
4667:
4663:
4657:
4652:
4648:
4643:
4637:
4631:
4627:
4623:
4620:
4613:
4608:
4605:
4601:
4597:
4592:
4588:
4584:
4581:
4578:
4573:
4570:
4566:
4509:
4506:
4479:
4476:
4473:
4470:
4467:
4447:
4444:
4424:
4404:
4401:
4398:
4378:
4375:
4372:
4369:
4366:
4346:
4343:
4340:
4337:
4334:
4331:
4326:
4322:
4310:
4309:
4294:
4290:
4286:
4283:
4280:
4276:
4272:
4267:
4264:
4260:
4256:
4253:
4246:
4242:
4238:
4235:
4232:
4228:
4224:
4219:
4216:
4212:
4206:
4201:
4197:
4167:
4162:
4158:
4146:
4145:
4134:
4131:
4128:
4125:
4122:
4108:
4107:
4095:
4089:
4084:
4081:
4077:
4073:
4068:
4065:
4062:
4057:
4054:
4050:
4045:
4040:
4034:
4029:
4025:
4021:
4016:
4011:
4007:
4002:
3993:
3989:
3985:
3982:
3979:
3976:
3972:
3967:
3961:
3958:
3951:
3946:
3943:
3939:
3935:
3930:
3927:
3924:
3919:
3916:
3912:
3894:
3893:
3882:
3877:
3874:
3870:
3866:
3861:
3858:
3854:
3850:
3847:
3843:
3834:
3830:
3826:
3821:
3816:
3812:
3805:
3797:
3793:
3789:
3784:
3779:
3775:
3767:
3763:
3757:
3754:
3749:
3746:
3709:
3706:
3700:
3697:
3684:
3664:
3644:
3622:
3618:
3614:
3611:
3608:
3586:
3582:
3578:
3558:
3538:
3535:
3532:
3529:
3526:
3504:
3501:
3498:
3493:
3488:
3483:
3479:
3475:
3470:
3467:
3464:
3459:
3454:
3449:
3445:
3422:
3417:
3413:
3407:
3403:
3399:
3396:
3376:
3356:
3336:
3324:
3321:
3305:
3302:
3298:
3277:
3255:
3252:
3248:
3227:
3206:
3186:
3166:
3163:
3160:
3157:
3137:
3134:
3131:
3128:
3116:
3113:
3100:
3080:
3060:
3057:
3054:
3051:
3048:
3028:
3000:
2997:
2994:
2991:
2988:
2966:
2959:
2955:
2952:
2949:
2946:
2942:
2936:
2933:
2930:
2926:
2919:
2915:
2912:
2909:
2906:
2902:
2896:
2893:
2890:
2886:
2877:
2873:
2869:
2866:
2863:
2860:
2857:
2854:
2849:
2845:
2822:
2819:
2814:
2810:
2789:
2786:
2781:
2777:
2756:
2736:
2705:
2685:
2682:
2677:
2673:
2669:
2666:
2663:
2660:
2637:
2634:
2631:
2628:
2625:
2605:
2602:
2599:
2596:
2593:
2573:
2570:
2567:
2564:
2561:
2558:
2553:
2549:
2528:
2525:
2522:
2519:
2516:
2513:
2508:
2504:
2491:
2488:
2473:
2469:
2463:
2460:
2456:
2450:
2446:
2442:
2437:
2433:
2429:
2424:
2420:
2416:
2411:
2407:
2403:
2398:
2394:
2390:
2387:
2382:
2378:
2372:
2369:
2365:
2361:
2358:
2353:
2349:
2345:
2340:
2336:
2332:
2329:
2326:
2321:
2317:
2313:
2289:
2284:
2280:
2274:
2270:
2264:
2260:
2256:
2253:
2231:
2226:
2222:
2216:
2212:
2206:
2202:
2198:
2195:
2168:
2148:
2126:
2122:
2116:
2113:
2109:
2105:
2102:
2097:
2093:
2089:
2084:
2080:
2076:
2073:
2070:
2065:
2061:
2057:
2035:
2031:
2005:
2000:
1997:
1994:
1990:
1986:
1981:
1977:
1973:
1968:
1964:
1960:
1957:
1931:
1927:
1923:
1920:
1898:
1892:
1889:
1884:
1880:
1876:
1873:
1870:
1865:
1862:
1857:
1853:
1849:
1846:
1843:
1835:
1830:
1827:
1824:
1820:
1816:
1813:
1810:
1807:
1802:
1798:
1793:
1788:
1784:
1778:
1775:
1771:
1767:
1764:
1759:
1755:
1751:
1746:
1742:
1738:
1735:
1732:
1727:
1723:
1719:
1716:
1708:
1704:
1700:
1697:
1691:
1686:
1680:
1677:
1674:
1670:
1666:
1663:
1659:
1633:
1630:
1625:
1622:
1619:
1615:
1592:
1586:
1583:
1578:
1574:
1570:
1567:
1564:
1559:
1556:
1551:
1547:
1543:
1540:
1537:
1529:
1524:
1521:
1518:
1514:
1509:
1503:
1500:
1497:
1493:
1489:
1486:
1482:
1475:
1472:
1467:
1463:
1459:
1456:
1453:
1448:
1445:
1440:
1436:
1432:
1429:
1426:
1418:
1413:
1410:
1407:
1403:
1399:
1394:
1391:
1388:
1384:
1380:
1377:
1355:
1343:
1340:
1323:
1303:
1279:
1259:
1235:
1215:
1212:
1207:
1203:
1199:
1196:
1193:
1190:
1166:
1146:
1126:
1123:
1120:
1117:
1097:
1094:
1091:
1088:
1068:
1044:
1041:
1038:
1035:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
965:
962:
959:
956:
953:
950:
947:
924:
904:
880:
875:
871:
867:
862:
858:
854:
849:
827:
824:
821:
818:
815:
812:
809:
787:
784:
781:
776:
771:
768:
746:
743:
740:
735:
730:
727:
715:
712:
697:
692:
689:
686:
682:
678:
673:
669:
665:
660:
656:
652:
649:
629:
607:
604:
601:
597:
576:
556:
551:
548:
545:
541:
537:
532:
529:
526:
522:
518:
494:
491:
486:
483:
479:
475:
472:
469:
466:
462:
457:
453:
448:
445:
442:
438:
434:
431:
427:
423:
419:
415:
410:
407:
404:
400:
396:
393:
389:
383:
380:
377:
374:
371:
367:
344:
341:
338:
335:
332:
328:
303:
300:
297:
293:
289:
284:
281:
278:
274:
270:
267:
263:
257:
254:
251:
247:
243:
238:
235:
231:
227:
224:
221:
218:
214:
209:
205:
200:
197:
194:
190:
186:
183:
179:
156:
153:
149:
145:
142:
139:
136:
132:
107:
104:
101:
98:
95:
92:
89:
77:
74:
72:
69:
41:systems theory
9:
6:
4:
3:
2:
7060:
7049:
7046:
7044:
7041:
7040:
7038:
7023:
7020:
7018:
7015:
7013:
7010:
7008:
7005:
7001:
6998:
6997:
6996:
6993:
6991:
6988:
6986:
6983:
6981:
6978:
6976:
6973:
6971:
6968:
6964:
6961:
6960:
6959:
6956:
6954:
6951:
6949:
6946:
6944:
6941:
6939:
6936:
6933:
6930:
6928:
6925:
6924:
6922:
6918:
6912:
6909:
6906:
6903:
6900:
6897:
6894:
6891:
6889:
6886:
6884:
6881:
6879:
6876:
6872:
6869:
6867:
6864:
6863:
6862:
6859:
6857:
6854:
6852:
6849:
6848:
6846:
6844:
6840:
6833:
6830:
6827:
6824:
6821:
6818:
6815:
6812:
6809:
6806:
6805:
6803:
6801:
6797:
6790:
6787:
6785:
6782:
6779:
6776:
6774:
6771:
6768:
6765:
6762:
6759:
6756:
6753:
6751:
6748:
6747:
6745:
6743:
6739:
6732:
6729:
6726:
6723:
6721:
6718:
6715:
6712:
6709:
6706:
6704:
6701:
6699:
6696:
6695:
6693:
6691:
6690:Finite volume
6687:
6676:
6673:
6670:
6667:
6664:
6661:
6660:
6658:
6654:
6648:
6645:
6643:
6640:
6638:
6635:
6633:
6630:
6628:
6625:
6624:
6622:
6620:
6616:
6610:
6607:
6604:
6601:
6600:
6598:
6596:
6592:
6589:
6587:
6583:
6579:
6572:
6567:
6565:
6560:
6558:
6553:
6552:
6549:
6538:
6534:
6529:
6524:
6520:
6516:
6513:(1): 93–108.
6512:
6508:
6501:
6492:
6487:
6483:
6479:
6475:
6471:
6467:
6460:
6452:
6448:
6444:
6440:
6436:
6432:
6428:
6424:
6417:
6410:
6402:
6398:
6394:
6390:
6386:
6382:
6378:
6374:
6367:
6360:
6352:
6348:
6344:
6340:
6336:
6332:
6328:
6324:
6319:
6314:
6310:
6306:
6299:
6292:
6290:
6281:
6277:
6273:
6269:
6265:
6261:
6257:
6253:
6246:
6239:
6231:
6227:
6223:
6219:
6215:
6211:
6204:
6196:
6190:
6186:
6179:
6171:
6167:
6163:
6159:
6155:
6151:
6147:
6143:
6139:
6132:
6124:
6120:
6116:
6112:
6108:
6104:
6100:
6096:
6089:
6081:
6077:
6073:
6069:
6050:
6042:
6038:
6034:
6030:
6026:
6022:
6015:
6007:
6003:
5999:
5995:
5991:
5987:
5983:
5979:
5972:
5970:
5961:
5957:
5952:
5947:
5943:
5939:
5935:
5928:
5920:
5916:
5911:
5906:
5902:
5898:
5894:
5887:
5879:
5875:
5868:
5860:
5856:
5852:
5848:
5844:
5840:
5833:
5825:
5819:
5811:
5807:
5803:
5801:9783662033296
5797:
5793:
5786:
5778:
5774:
5770:
5766:
5762:
5758:
5751:
5743:
5739:
5735:
5731:
5724:
5722:
5720:
5711:
5707:
5703:
5697:
5693:
5686:
5678:
5674:
5670:
5666:
5662:
5658:
5653:
5648:
5644:
5640:
5633:
5631:
5629:
5627:
5625:
5616:
5612:
5608:
5604:
5600:
5596:
5589:
5587:
5585:
5576:
5572:
5567:
5562:
5558:
5554:
5550:
5543:
5529:on 2011-08-11
5528:
5524:
5518:
5514:
5510:
5502:
5494:
5490:
5485:
5480:
5476:
5472:
5468:
5464:
5457:
5455:
5446:
5442:
5437:
5432:
5428:
5424:
5420:
5416:
5412:
5405:
5403:
5401:
5399:
5390:
5386:
5382:
5378:
5374:
5370:
5366:
5360:
5358:
5349:
5345:
5341:
5337:
5332:
5327:
5323:
5319:
5315:
5311:
5304:
5302:
5300:
5298:
5296:
5291:
5283:
5281:
5270:
5256:
5237:
5233:
5223:
5213:
5208:
5204:
5178:
5174:
5164:
5154:
5149:
5145:
5136:
5118:
5114:
5091:
5087:
5063:
5060:
5055:
5051:
5047:
5044:
5039:
5035:
5031:
5025:
5022:
5012:
5011:
5010:
5008:
4998:
4994:
4992:
4988:
4964:
4960:
4951:
4945:
4942:
4939:
4934:
4931:
4927:
4921:
4916:
4912:
4908:
4903:
4899:
4895:
4892:
4889:
4884:
4881:
4877:
4871:
4866:
4862:
4857:
4851:
4845:
4841:
4837:
4827:
4823:
4819:
4816:
4813:
4808:
4805:
4801:
4797:
4792:
4789:
4786:
4781:
4778:
4774:
4763:
4762:
4742:
4738:
4729:
4723:
4718:
4715:
4711:
4705:
4700:
4696:
4692:
4687:
4683:
4679:
4676:
4673:
4668:
4665:
4661:
4655:
4650:
4646:
4641:
4635:
4629:
4625:
4621:
4611:
4606:
4603:
4599:
4595:
4590:
4586:
4582:
4579:
4576:
4571:
4568:
4564:
4553:
4552:
4551:
4549:
4545:
4540:
4538:
4534:
4530:
4526:
4521:
4507:
4496:
4491:
4474:
4471:
4468:
4445:
4442:
4422:
4402:
4399:
4396:
4373:
4370:
4367:
4341:
4338:
4335:
4329:
4324:
4320:
4292:
4288:
4284:
4281:
4278:
4274:
4270:
4265:
4262:
4258:
4254:
4251:
4244:
4240:
4236:
4233:
4230:
4226:
4222:
4217:
4214:
4210:
4204:
4199:
4195:
4187:
4186:
4185:
4183:
4165:
4160:
4156:
4132:
4126:
4123:
4113:
4112:
4111:
4093:
4087:
4082:
4079:
4075:
4071:
4066:
4063:
4060:
4055:
4052:
4048:
4043:
4038:
4032:
4027:
4023:
4019:
4014:
4009:
4005:
4000:
3991:
3983:
3974:
3970:
3965:
3959:
3949:
3944:
3941:
3937:
3933:
3928:
3925:
3922:
3917:
3914:
3910:
3899:
3898:
3897:
3875:
3872:
3868:
3864:
3859:
3856:
3852:
3845:
3841:
3832:
3828:
3819:
3814:
3803:
3795:
3791:
3782:
3777:
3765:
3761:
3755:
3747:
3734:
3733:
3732:
3729:
3727:
3723:
3714:
3705:
3696:
3682:
3662:
3642:
3620:
3616:
3612:
3609:
3606:
3584:
3580:
3576:
3556:
3536:
3533:
3530:
3527:
3524:
3502:
3499:
3496:
3486:
3481:
3477:
3473:
3468:
3465:
3462:
3452:
3447:
3443:
3420:
3415:
3411:
3405:
3401:
3397:
3394:
3374:
3354:
3334:
3320:
3303:
3300:
3296:
3275:
3253:
3250:
3246:
3225:
3204:
3184:
3161:
3155:
3132:
3126:
3112:
3098:
3078:
3058:
3055:
3052:
3049:
3046:
3026:
3018:
3014:
2995:
2992:
2989:
2977:
2964:
2950:
2944:
2934:
2931:
2928:
2910:
2904:
2894:
2891:
2888:
2875:
2867:
2861:
2858:
2855:
2847:
2843:
2834:
2820:
2817:
2812:
2787:
2784:
2779:
2754:
2734:
2725:
2723:
2719:
2703:
2683:
2680:
2675:
2671:
2667:
2664:
2661:
2658:
2651:
2632:
2629:
2626:
2600:
2597:
2594:
2568:
2565:
2562:
2556:
2551:
2523:
2520:
2517:
2511:
2506:
2487:
2471:
2461:
2458:
2448:
2435:
2431:
2422:
2409:
2396:
2392:
2385:
2380:
2370:
2367:
2359:
2351:
2347:
2338:
2327:
2319:
2315:
2303:
2287:
2282:
2278:
2272:
2262:
2258:
2254:
2251:
2229:
2224:
2220:
2214:
2204:
2200:
2196:
2193:
2186:
2182:
2166:
2146:
2124:
2114:
2111:
2103:
2095:
2091:
2082:
2071:
2063:
2059:
2033:
2029:
2021:
2003:
1998:
1995:
1992:
1979:
1975:
1971:
1966:
1962:
1947:
1946:operator norm
1929:
1921:
1909:
1896:
1887:
1882:
1878:
1874:
1871:
1860:
1855:
1851:
1847:
1844:
1833:
1828:
1825:
1822:
1818:
1814:
1808:
1800:
1796:
1791:
1786:
1776:
1773:
1765:
1757:
1753:
1744:
1733:
1725:
1721:
1714:
1706:
1698:
1689:
1675:
1668:
1664:
1661:
1645:
1631:
1628:
1620:
1613:
1603:
1590:
1581:
1576:
1572:
1568:
1565:
1554:
1549:
1545:
1541:
1538:
1527:
1522:
1519:
1516:
1512:
1507:
1498:
1491:
1487:
1484:
1480:
1470:
1465:
1461:
1457:
1454:
1443:
1438:
1434:
1430:
1427:
1416:
1411:
1408:
1405:
1401:
1397:
1389:
1382:
1378:
1375:
1367:
1353:
1339:
1337:
1321:
1301:
1291:
1277:
1257:
1249:
1233:
1213:
1210:
1205:
1201:
1197:
1194:
1191:
1188:
1180:
1164:
1144:
1121:
1115:
1092:
1086:
1066:
1058:
1039:
1033:
1010:
1004:
998:
995:
989:
983:
963:
960:
957:
954:
951:
948:
945:
938:The equation
936:
922:
902:
894:
873:
869:
865:
860:
856:
825:
822:
819:
816:
813:
810:
807:
785:
782:
779:
769:
766:
744:
741:
738:
728:
725:
711:
695:
690:
687:
684:
671:
667:
663:
658:
654:
627:
602:
595:
574:
549:
546:
543:
539:
535:
530:
527:
524:
520:
506:
492:
489:
481:
477:
473:
470:
467:
460:
455:
451:
446:
443:
440:
436:
432:
429:
425:
421:
417:
413:
408:
405:
402:
398:
394:
391:
387:
378:
375:
372:
365:
339:
336:
333:
326:
318:2. Solve for
314:
301:
298:
295:
291:
287:
282:
279:
276:
272:
268:
265:
261:
252:
245:
241:
233:
229:
225:
222:
219:
212:
207:
203:
198:
195:
192:
188:
184:
181:
177:
151:
147:
143:
140:
137:
130:
122:1. Solve for
119:
105:
102:
99:
96:
93:
90:
87:
68:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
19:
6814:Peridynamics
6632:Lax–Wendroff
6528:10356/137201
6510:
6506:
6500:
6465:
6459:
6426:
6422:
6409:
6393:10356/138296
6379:(2): 85–87.
6376:
6372:
6359:
6343:10356/138249
6308:
6304:
6272:10356/138245
6255:
6251:
6238:
6213:
6209:
6203:
6184:
6178:
6148:(1): 69–84,
6145:
6141:
6131:
6101:(1): 41–63,
6098:
6094:
6088:
6071:
6067:
6049:
6024:
6020:
6014:
5981:
5977:
5941:
5937:
5927:
5900:
5896:
5886:
5877:
5873:
5867:
5842:
5838:
5832:
5791:
5785:
5760:
5756:
5750:
5733:
5729:
5691:
5685:
5642:
5638:
5601:(9): 43–58.
5598:
5594:
5556:
5552:
5542:
5531:. Retrieved
5527:the original
5512:
5501:
5469:(1): 28–41,
5466:
5462:
5418:
5414:
5372:
5368:
5331:11585/586011
5313:
5309:
5276:
5267:
5078:
5004:
4995:
4984:
4547:
4543:
4541:
4522:
4492:
4311:
4181:
4147:
4109:
3895:
3730:
3719:
3702:
3326:
3323:Factored ADI
3118:
3019:in 1877 for
2978:
2835:
2726:
2493:
2304:
1910:
1646:
1604:
1368:
1345:
1292:
937:
717:
509:The numbers
508:
317:
121:
79:
32:
26:
6948:Collocation
6491:10220/20410
5310:SIAM Review
7037:Categories
6637:MacCormack
6619:Hyperbolic
6318:2011.14043
6258:(1): 7–9.
5794:. Berlin.
5742:1911/20641
5736:(Thesis).
5701:1421407949
5652:1609.09494
5533:2011-08-18
5286:References
3177:, or when
2018:defines a
891:using the
76:The method
6953:Level-set
6943:Multigrid
6893:Balancing
6595:Parabolic
6170:1040-7790
6123:121455963
6115:0029-599X
6074:: 42–65,
6041:0036-1399
6006:1064-8275
5960:0377-0427
5919:0021-9045
5818:cite book
5810:883382758
5777:0167-6911
5710:824733531
5669:0895-4798
5615:0898-1221
5575:0898-1221
5445:0377-0427
5389:0895-4798
5340:0036-1445
5230:∂
5220:∂
5171:∂
5161:∂
5026:˙
4987:symmetric
4957:Δ
4913:δ
4863:δ
4835:Δ
4798:−
4735:Δ
4697:δ
4647:δ
4619:Δ
4596:−
4505:Δ
4285:−
4252:−
4196:δ
4157:δ
4130:Δ
4121:Δ
4024:δ
4006:δ
3981:Δ
3957:Δ
3934:−
3825:∂
3811:∂
3788:∂
3774:∂
3753:∂
3745:∂
3621:∗
3610:≈
3585:∗
3500:×
3487:∈
3466:×
3453:∈
3421:∗
3301:−
3251:−
3156:σ
3127:σ
3053:−
3017:Zolotarev
2932:∈
2892:∈
2818:⊂
2809:Λ
2785:⊂
2776:Λ
2767:, where
2676:∗
2557:⊂
2548:Λ
2512:⊂
2503:Λ
2468:‖
2459:−
2445:Λ
2428:‖
2419:‖
2406:Λ
2389:‖
2377:‖
2368:−
2344:‖
2335:‖
2312:‖
2288:∗
2269:Λ
2230:∗
2211:Λ
2183:and have
2121:‖
2112:−
2088:‖
2079:‖
2056:‖
1976:β
1963:α
1926:‖
1922:⋅
1919:‖
1879:β
1875:−
1852:α
1848:−
1819:∏
1783:‖
1774:−
1750:‖
1741:‖
1718:‖
1715:≤
1703:‖
1696:‖
1665:−
1605:Choosing
1573:α
1569:−
1546:β
1542:−
1513:∏
1488:−
1462:β
1458:−
1435:α
1431:−
1402:∏
1379:−
1206:∗
1116:σ
1087:σ
1034:σ
1014:∅
999:σ
996:∩
984:σ
952:−
814:−
783:×
770:∈
742:×
729:∈
668:β
655:α
540:β
521:α
490:−
437:α
433:−
399:α
395:−
357:, where
273:β
269:−
189:β
185:−
94:−
49:parabolic
6927:Spectral
6866:additive
6789:Smoothed
6755:Extended
6537:61039449
6401:22940993
6351:37135325
6280:29025478
5348:17271167
3599:, where
3435:, where
3039:= and
2584:, where
1685:‖
1658:‖
1057:spectrum
1026:, where
800:, then
169:, where
67:method.
53:elliptic
6911:FETI-DP
6791:(S-FEM)
6710:(MUSCL)
6698:Godunov
6470:Bibcode
6451:7578037
6431:Bibcode
6323:Bibcode
6218:Bibcode
6150:Bibcode
6080:0071875
5986:Bibcode
5880:: 1–59.
5847:Bibcode
5677:3828461
5493:0071874
5423:Bibcode
4110:where:
3517:, with
2302:, then
1944:is the
1055:is the
45:control
6920:Others
6907:(FETI)
6901:(BDDC)
6773:Mortar
6757:(XFEM)
6750:hp-FEM
6733:(WENO)
6716:(AUSM)
6677:(FDTD)
6671:(FDFD)
6656:Others
6642:Upwind
6605:(FTCS)
6535:
6449:
6399:
6349:
6278:
6191:
6168:
6121:
6113:
6078:
6039:
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5675:
5667:
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5387:
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5338:
5135:Banach
5079:where
4525:banded
3288:, and
1911:where
31:, the
6934:(DVR)
6895:(BDD)
6834:(PIC)
6828:(MPM)
6822:(MPS)
6810:(SPH)
6780:(GDM)
6769:(SEM)
6727:(ENO)
6665:(ADI)
6533:S2CID
6447:S2CID
6419:(PDF)
6397:S2CID
6369:(PDF)
6347:S2CID
6313:arXiv
6301:(PDF)
6276:S2CID
6248:(PDF)
6119:S2CID
5673:S2CID
5647:arXiv
5344:S2CID
4312:with
2139:. If
1226:when
759:and
6816:(PD)
6763:(DG)
6189:ISBN
6166:ISSN
6111:ISSN
6037:ISSN
6002:ISSN
5956:ISSN
5915:ISSN
5824:link
5806:OCLC
5796:ISBN
5773:ISSN
5706:OCLC
5696:ISBN
5665:ISSN
5611:ISSN
5571:ISSN
5517:ISBN
5441:ISSN
5385:ISSN
5336:ISSN
5196:and
5106:and
4389:for
4148:and
3675:and
3347:and
3148:and
3091:and
2800:and
2747:and
2616:and
2539:and
2244:and
2179:are
2159:and
1270:and
1177:are
1157:and
1108:and
915:and
51:and
43:and
6523:hdl
6515:doi
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6478:doi
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6062:= u
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2604:]
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