High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources

Journal of Computational Physics

Volumn 213, Issue 1, 2006, Pages 1-30

  Zhou, Y C ^a   Zhao, Shan ^a   Feig, Michael ^a,b   Wei, G W ^a,c  

^a MICHIGAN STATE UNIVERSITY   (United States)

^b Michigan State University   (United States)

^c Michigan State University   (United States)

Author keywords

Elliptic equations; Ghost fluid method; High order method; Immersed boundary method; Immersed interface method; Matched interface and boundary

Indexed keywords

DIFFERENTIAL EQUATIONS; TURBULENT FLOW;

BOUNDARY METHOD; ELLIPTIC EQUATIONS; GHOST FLUID METHOD; HIGH-ORDER; HIGH-ORDER METHODS; HIGHER-ORDER; HIGHER-ORDER METHODS; IMMERSED BOUNDARY METHODS; IMMERSED INTERFACE METHOD; MATCHED INTERFACE AND BOUNDARY;

FINITE DIFFERENCE METHOD;

EID: 31744448526     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2005.07.022     Document Type: Article

References (55)

1. L. Adams Z.L. Li The immersed interface/multigrid methods for interface problems SIAM J. Sci. Comput. 24 2002 463-479
2. L. Adams T.P. Chartier New geometric immersed interface multigrid solvers SIAM J. Sci. Comput. 25 2004 1516-1533
3. I. Babuška The finite element method for elliptic equations with discontinuous coefficients Computing 5 1970 207-213
4. H. Ben Ameur M. Burger B. Hackl Level set methods for geometric inverse problems in linear elasticity Inverse Problems 20 2004 673-696
5. P.A. Berthelsen A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions J. Comput. Phys. 197 2004 364-386
6. G. Biros L.X. Ying D. Zorin A fast solver for the Stokes equations with distributed forces in complex geometries J. Comput. Phys. 193 2004 317-348
7. J. Bramble J. King A finite element method for interface problems in domains with smooth boundaries and interfaces Adv. Comput. Math. 6 1996 109-138
8. Z. Chen J. Zou Finite element methods and their convergence for elliptic and parabolic interface problems Numer. Math. 79 1998 175-202
9. S.Z. Deng K. Ito Z.L. Li Three-dimensional elliptic solvers for interface problems and applications J. Comput. Phys. 184 2003 215-243
10. M.A. Dumett J.P. Keener An immersed interface method for solving anisotropic elliptic boundary value problems in three dimensions SIAM J. Sci. Comput. 25 2003 348-367
11. E.A. Fadlun R. Verzicco P. Orlandi J. Mohd-Yusof Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations J. Comput. Phys. 161 2000 30-60
12. R.P. Fedkiw T. Aslam B. Merriman S. Osher A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method) J. Comput. Phys. 152 1999 457-492
13. M. Feig C.L. Brooks III Recent advances in the development and application of implicit solvent models in biomolecule simulations Curr. Opin. Struct. Bio. 14 2004 217-224
14. B. Fornberg Calculation of weights in finite difference formulas SIAM Rev. 40 1998 685-691
15. F. Gibou R.P. Fedkiw A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem J. Comput. Phys. 202 2005 577-601
16. S. Hou X.-D. Liu A numerical method for solving variable coefficient elliptic equation with interfaces J. Comput. Phys. 202 2005 411-445
17. T.Y. Hou Z.L. Li S. Osher H. Zhao A hybrid method for moving interface problems with application to the Hele-Shaw flow J. Comput. Phys. 134 1997 236-252
18. H. Huang Z.L. Li Convergence analysis of the immersed interface method IMA J. Numer. Anal. 19 1999 583-608
19. J.K. Hunter Z.L. Li H. Zhao Reactive autophobic spreading of drops J. Comput. Phys. 183 2002 335-366
20. G. Iaccarino R. Verzicco Immersed boundary technique for turbulent flow simulations Appl. Mech. Rev. 56 2003 331-347
21. S. Jin X.L. Wang Robust numerical simulation of porosity evolution in chemical vapor infiltration II. Two-dimensional anisotropic fronts J. Comput. Phys. 179 2002 557-577
22. H. Johansen P. Colella A Cartesian grid embedding boundary method for Poisson's equation on irregular domains J. Comput. Phys. 147 1998 60-85
23. J.D. Kandilarov Immersed interface method for a reaction-diffusion equation with a moving own concentrated source Lecture Notes Comput. Sci. 2542 2003 506-513
24. L. Lee R.J. LeVeque An immersed interface method for incompressible Navier-Stokes equations SIAM J. Sci. Comput. 25 2003 832-856
25. R.J. LeVeque Z.L. Li The immersed interface method for elliptic equations with discontinuous coefficients and singular sources SIAM J. Numer. Anal. 31 1994 1019-1044
26. Z.L. Li A fast iterative algorithm for elliptic interface problems SIAM J. Numer. Anal. 35 1998 230-254
27. Z.L. Li The immersed interface method using a finite element formulation Appl. Numer. Math. 27 1998 253-267
28. Z.L. Li An overview of the immersed interface method and its applications Taiwanese J. Math. 7 2003 1-49
29. Z.L. Li K. Ito Maximum principle preserving schemes for interface problems with discontinuous coefficients SIAM J. Sci. Comput. 23 2001 339-361
30. Z.L. Li S.R. Lubkin Numerical analysis of interfacial two-dimensional Stokes flow with discontinuous viscosity and variable surface tension Int. J. Numer. Meth. Fluid. 37 2001 525-540
31. Z.L. Li W.-C. Wang I.-L. Chern M.-C. Lai New formulations for interface problems in polar coordinates SIAM J. Sci. Comput. 25 2003 224-245
32. S. Lim C.S. Peskin Simulations of the whirling instability by the immersed boundary method SIAM J. Sci. Comput. 25 2004 2066-2083
33. M.N. Linnick H.F. Fasel A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains J. Comput. Phys. 204 2005 157-192
34. X.-D. Liu S. Osher T. Chan Weighted essentially non-oscillatory schemes J. Comput. Phys. 115 1994 200-212
35. X.D. Liu R.P. Fedkiw M. Kang A boundary condition capturing method for Poisson's equation on irregular domains J. Comput. Phys. 160 2000 151-178
36. B. Lombard J. Piraux How to incorporate the spring-mass conditions in finite-difference schemes SIAM J. Sci. Comput. 24 2003 1379-1407
37. A. Mayo The fast solution of Poisson's and the biharmonic equations on irregular regions SIAM J. Numer. Anal. 21 1984 285-299
38. A. Mckenney L. Greengard A. Mayo A fast Poisson solver for complex geometries J. Comput. Phys. 118 1995 348-355
39. C.S. Peskin Numerical analysis of blood flow in heart J. Comput. Phys. 25 1977 220-252
40. C.S. Peskin Lectures on mathematical aspects of physiology Lect. Appl. Math. 19 1981 69-107
41. W.H. Press S.A. Teukolsky W.T. Vetterling B.P. Flannery Numerical Recipes in Fortran: The Art of Scientific Computing 1999 Cambridge University Press Cambridge
42. M. Schulz G. Steinebach Two-dimensional modelling of the river Rhine J. Comput. Appl. Math. 145 2002 11-20
43. J.A. Sethian A. Wiegmann Structural boundary design via level set and immersed interface methods J. Comput. Phys. 163 2000 489-528
44. A.K. Tornberg B. Engquist Numerical approximations of singular source terms in differential equations J. Comput. Phys. 200 2004 462-488
45. Y.-H. Tseng J.H. Ferziger A ghost-cell immersed boundary method for flow in complex geometry J. Comput. Phys. 192 2003 593-623
46. J.V. Voorde J. Vierendeels E. Dick Flow simulations in rotary volumetric pumps and compressors with the fictitious domain method J. Comput. Appl. Math. 168 2004 491-499
47. J.H. Walther G. Morgenthal An immersed interface method for the vortex-in-cell algorithm J. Turbulence 3 039 2002
48. W.-C. Wang A jump condition capturing finite difference scheme for elliptic interface problems SIAM J. Sci. Comput. 25 2004 1479-1496
49. G.W. Wei Discrete singular convolution for the solution of the Fokker-Planck equations J. Chem. Phys. 110 1999 8930-8942
50. G.W. Wei Y. Gu Conjugated filter approach for solving Burgers' equation J. Comput. Appl. Math. 149 2002 439-456
51. A. Wiegmann K.P. Bube The explicit-jump immersed interface method: finite difference methods for PDEs with piecewise smooth solutions SIAM J. Numer. Anal. 37 2000 827-862
52. S. Zhao G.W. Wei High order FDTD methods via derivative matching for Maxwell's equations with material interfaces J. Comput. Phys. 200 2004 60-103
53. S. Zhao G.W. Wei Tensor product derivative matching for wave propagation in inhomogeneous media Microwave and Optical Technology Letters 43 2004 69-77
54. S. Zhao G.W. Wei Y. Xiang DSC analysis of free-edged beams by an iteratively matched boundary method J Sound Vibrat 284 2005 487-493
55. Y.C. Zhou G.W. Wei High-resolution conjugate filters for the simulation of flows J. Comput. Phys. 189 2003 150-179