Adaptive error control for steady state solutions of inviscid flow

SIAM Journal on Scientific Computing

Volumn 23, Issue 5, 2002, Pages 1777-1798

  Ferm, Lars ^a   Lötstedt, Per ^a  

^a UPPSALA UNIVERSITY   (Sweden)

Author keywords

Discretization error; Error control; Error equation; Euler equations; Finite volume method

Indexed keywords

ALGORITHMS; ERROR ANALYSIS; FINITE VOLUME METHOD;

DISCRETIZATION ERRORS;

COMPUTATIONAL FLUID DYNAMICS;

EID: 0036982847     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827500367452     Document Type: Article

References (43)

1. R. ABGRALL, On essentially non-oscillatory schemes on unstructured meshes: Analysis and implementation, J. Comput. Phys., 114 (1994), pp. 45-58.
2. J. D. ANDERSON, JR., Modern Compressible Flow, 2nd ed., McGraw-Hill, New York, 1990.
3. J. BELL, M. BERGER, J. SALTZMAN, AND M. WELCOME, Three-dimensional adaptive mesh refinement for hyperbolic conservation laws, SIAM J. Sci. Comput., 15 (1994), pp. 127-138.
4. M. BERGER AND P. COLELLA, Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82 (1989), pp. 64-84.
5. M. BERGER AND A. JAMESON, Automatic adaptive grid refinement for the Euler equations, AIAA J., 23 (1985), pp. 561-568.
6. C. J. BUDD, G. P. KOOMULLIL, AND A. M. STUART, On the solution of convection-diffusion boundary value problems using equidistributed grids, SIAM J. Sci. Comput., 20 (1998), pp. 591-618.
7. J. CASPER AND H. L. ATKINS, A finite-volume high-order ENO scheme for two-dimensional hyperbolic systems, J. Comput. Phys., 106 (1993), pp. 62-76.
8. A. J. CHORIN, A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2 (1967), pp. 12-36.
9. S. D. CONNELL AND D. G. HOLMES, Three-dimensional unstructured adaptive multigrid scheme for the Euler equations, AIAA J., 32 (1994), pp. 1626 1632.
10. P. COLELLA, M. R. DORR, AND D. D. WAKE, Numerical solution of plasma fluid equations using locally refined grids, J. Comput. Phys., 152 (1999), pp. 550-583.
11. A. DADONE AND B. GROSSMAN, Surface boundary conditions for the numerical solution of the Euler equations, AIAA J., 32 (1994), pp. 285-293.
12. R. L. DAVIS AND J. F. DANNENHOFFER III, 3-D Adaptive Grid-Embedding Euler Technique, AIAA paper 93-0330, AIAA, Washington, DC, 1993.
13. D. DE ZEEUW AND K. G. POWELL, An adaptively refined Cartesian mesh solver for the Euler equations, J. Comput. Phys., 104 (1993), pp. 56-68.
14. J. DE KEYSER, K. LUST, AND D. ROOSE, Run-time balancing support for a parallel multiblock Euler/Navier-Stokes code with adaptive refinement on distributed memory computers, Parallel Comput., 20 (1994), pp. 1069-1088.
15. G. EFRAIMSSON AND G. KREISS, A remark on numerical errors downstream of slightly viscous shocks, SIAM J. Numer. Anal., 36 (1999), pp. 853-863.
16. B. ENGQUIST AND B. SJÖGREEN, The convergence rate of finite difference schemes in the presence of shocks, SIAM J. Numer. Anal., 35 (1998), pp. 2464-2485.
17. K. ERIKSSON, D. ESTEP, P. HANSBO, AND C. JOHNSON, Introduction to adaptive methods for differential equations, Acta Numer., (1995), pp. 121-123.
18. L. FERM AND P. LÖTSTEDT, Blockwise adaptive grids with multigrid acceleration for compressible flow, AIAA J., 37 (1999), pp. 121-123.
19. L. FERM AND P. LÖTSTEDT, On numerical errors in the boundary conditions of the Euler equations, Appl. Math. Comput., to appear.
20. L. FERM AND P. LÖTSTEDT, Efficiency in the adaptive solution of inviscid compressible flow problems, Nonlinear Anal., 47 (2001), pp. 3467-3478.
21. M. GERRITSEN AND P. OLSSON, Designing an efficient solution strategy for fluid flows II. Stable high-order central finite difference schemes on composite adaptive grids with sharp shock resolution, J. Comput. Phys., 147 (1998), pp. 293-317.
22. B. GUSTAFSSON, The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comput., 29 (1975), pp. 396-406.
23. E. HAIRER, S. P. NØRSETT, AND G. WANNER, Solving Ordinary Differential Equations, 2nd ed., Springer-Verlag, Berlin, 1993.
24. R. D. HORNUNG AND J. A. TRANGENSTEIN, Adaptive mesh refinement and multilevel iteration for flow in porous media, J. Comput. Phys., 136 (1997), pp. 522-545.
25. C. ILINCA, X. D. ZHANG, J.-Y. TRÉPANIER, AND R. CAMARERO, A comparison of three estimation techniques for finite-volume solutions of compressible flow, Comput. Methods Appl. Mech. Engrg., 189 (2000), pp. 1277-1294.
26. A. JAMESON, W. SCHMIDT, AND E. TURKEL, Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes, AIAA-paper 81-1259, AIAA, Reston, WA, 1981.
27. S. LI, L. PETZOLD, AND Y. REN, Stability of moving mesh systems of partial differential equations, SIAM J. Sci. Comput., 20 (1998), pp. 719-738.
28. F. LIU, S. JI, AND G. LIAO, An adaptive grid method and its application to steady Euler flow calculations, SIAM J. Sci. Comput., 20 (1998), pp. 811-825.
29. P. LÖTSTEDT, S. SÖDERBERG, A. RAMAGE, AND L. HEMMINGSSON-FRÄNDÉN, Implicit solution of hyperbolic equations with space-time adaptivity, BIT, 42 (2002), pp. 134-158.
30. S. MUZAFERIJA AND D. GOSMAN, Finite-volume CFD procedure and adaptive error control strategy for grids of arbitrary topology, J. Comput. Phys., 138 (1997), pp. 776-787.
31. W. L. OBERKAMPF AND F. G. BLOTTNER, Issues in computational fluid dynamics code verification and validation, AIAA J., 36 (1998), pp. 687-695.
32. S. OSHER AND F. SOLOMON, Upwind schemes for hyperbolic systems of conservation laws, Math. Comp., 38 (1982), pp. 339-377.
33. N. G. PANTELELIS AND A. E. KANARACHOS, The parallel block adaptive multigrid method for the implicit solution of the Euler equations, Internat. J. Numer. Methods Fluids, 22 (1996), pp. 411-428.
34. A. PAPASTAVROU AND R. VERFÜRTH, A posteriori error estimators for stationary convection-diffusion problems: A computational comparison, Comput. Methods Appl. Mech. Engrg., 189 (2000), pp. 449-462.
35. K. G. POWELL, A tree-based adaptive scheme for solution of the equations of gas dynamics and magnetohydrodynamics, Appl. Numer. Math., 14 (1994), pp. 327-352.
36. K. G. POWELL, P. L. ROE, T. J. LINDE, T. I. GOMBOSI, AND D. L. DE ZEEUW, A solution-adaptive upwind scheme for ideal magnetohydrodynamics, J. Comput. Phys., 154 (1999), pp. 284-309.
37. M. RENARDY AND R. C. ROGERS, An Introduction to Partial Differential Equations, Springer-Verlag, New York, 1993.
38. P. ROACHE, Quantification of uncertainty in computational fluid dynamics, Annu. Rev. Fluid Mech., 29 (1997), pp. 123-160.
39. G. SÖDERLIND, The automatic control of numerical integration, CWI Quarterly, 11 (1998), pp. 55-74.
40. K. SRINIVASAN AND S. G. RUBIN, Solution-based grid adaptation through segmented multigrid domain decomposition, J. Comput. Phys., 136 (1997), pp. 467-493.
41. D. A. VENDITTI AND D. L. DARMOFAL, Adjoint error estimation and grid adaptation for functional outputs: Application to quasi-one-dimensional flow, J. Comput. Phys., 164 (2000), pp. 204-227.
42. J. WU, H. RITZDORF, K. OOSTERLEE, B. STECKEL, AND A. SCHÜLLER, Adaptive parallel multigrid solution of 2D incompressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids, 24 (1997), pp. 875-892.
43. X. D. ZHANG, J.-Y. TRÉPANIER, AND R. CAMARERO, A posteriori error estimation for finite-volume solutions of hyperbolic conservation laws, Comput. Methods Appl. Mech. Engrg., 185 (2000), pp. 1-19.