ON the ANALYSIS of FINITE VOLUME METHODS for EVOLUTIONARY PROBLEMS

SIAM Journal on Numerical Analysis

Volumn 35, Issue 6, 1998, Pages 2195-2222

  Morton, K W ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Characteristic galerkin; Conservation laws; Error decomposition; Evolution galerkin; Finite volume; Godunov formulation; Lagrange galerkin; Recovery techniques

Indexed keywords

EID: 0000534145     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142997316967     Document Type: Article

References (44)

1. T.J. BARTH AND P.O. FREDERICKSON, Higher Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction, Paper-90-0013, AIAA, Washington, DC, 1990.
2. J.P. BENQUÉ. G. LABADIE, AND J. RONAT, A new finite element method for the Navier-Stokes equations coupled with a temperature equation, in Proceedings of the Fourth International Symposium on Finite Element Methods in Flow Problems, T. Kawai, ed., North-Holland, Amsterdam, 1982, pp. 295-302.
3. Y. BRENIER, Average multivalued solutions for scalar conservation laws, SIAM J. Numer. Anal., 21 (1984), pp. 1013-1037.
4. P.N. CHILDS AND K.W. MORTON, Characteristic Galerkin methods for scalar conservation laws in one dimension, SIAM J. Numer. Anal., 27 (1990), pp, 553-594.
5. P. COLELLA AND P. WOODWARD, The piecewise parabolic method (PPM) for gas-dynamical simulations, J. Comput. Phys., 54 (1984), pp. 174-201.
6. P.I. CRUMPTON AND M.B. GILES, Aircraft Computations Using Multigrid and an Unstructured Parallel Library. Paper 95-0210, AIAA, Washington, DC, 1995.
7. P.I. CRUMPTON, J.A. MACKENZIE, AND K.W. MORTON, Cell vertex algorithms for the compressible Navier-Stokes equations, J. Comput. Phys., 109 (1993), pp. 1-15.
8. T. DAVIES AND J.C.R. HUNT, New developments in numerical weather prediction, in Numerical Methods for Fluid Dynamics V, K.W. Morton and M.J. Baines, eds., Clarendon Press, Oxford, UK, 1995. pp. 1-18.
9. J. DONEA. A Taylor-Galerkin method for convective transport problems, Intern. J. Numer. Methods Engrg., 20 (1984), pp. 101-119.
10. J. DONEA, V. SELMIN, AND L. QUARTAPELLE, Recent developments of the Taylor-Galerkin method for the numerical solution of hyperbolic problems, in Numerical Methods for Fluid Dynamics III, K.W. Morton and M.J. Baines, eds., Clarendon Press, Oxford, UK, 1988, pp. 171-185.
11. J. DOUGLAS, JR. AND T.F. RUSSELL, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal., 19 (1982), pp. 871-885.
12. B. ENGQUIST AND S. OSHER, One-sided difference approximations for nonlinear conservation laws, Math. Comp., 36 (1981), pp. 321-352.
13. M. FEY, Ein echt mehrdimensionales Verfahren zur Lösung der Eulergleichungen, dissertation, ETH Zürich, Zurich, Switzerland, 1993.
14. E. GODLEWSKI AND P.-A. RAVIART, Hyperbolic Systems of Conservation Laws, Mathematics and Applications 3/4, Ellipses, Paris, 1991.
15. S.K. GODUNOV, A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb., 47 (1959), pp. 271-306.
16. A. HARTEN, High resolution schemes for conservation laws, J. Comput. Phys., 49 (1983), pp. 357-393.
17. A. HARTEN, On a class of high resolution total-variation-stable finite-difference schemes, SIAM J. Numer. Anal., 21 (1984), pp. 1-23.
18. A. HARTEN, B. ENGQUIST, S. OSHER, AND S.R. CHAKRAVARTHY, Uniformly high order accurate essentially non-oscillatory schemes, III, J. Comput. Phys., 71 (1987), pp. 231-303.
19. A. HARTEN, P. LAX, AND B. VAN LEER, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (1983), pp. 35-61.
20. B.P. LEONARD, A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Comput. Methods Appl. Mech. Engrg., 19 (1979), pp. 59-98.
21. B.P. LEONARD, The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection, Comput. Methods Appl. Mech. Engrg., 88 (1991), pp. 17-74.
22. B.P. LEONARD AND H.S. NIKNAFS, Sharp monotonic resolution of discontinuities without clipping of narrow extrema, Comput. & Fluids, 19 (1991), pp. 141-154.
23. P. LESAINT, Numerical solution of the equation of continuity, in Topics in Numerical Analysis III, J.J.H. Miller, ed., Academic Press, New York, 1977, pp. 199-222.
24. P. LIN, Godunov Type of the Euler Characteristic Galerkin Scheme for One-dimensional Scalar Conservation Laws, Tech. Report NA95/02, Oxford University Computing Laboratory, Oxford, UK, 1995.
25. P. LIN AND K.W. MORTON, An Upwind Finite Element Method for Nonlinear Hyperbolic Systems of Conservation Laws, Tech. Report NA93/10, Oxford University Computing Laboratory, Oxford, UK, 1993.
26. P. LIN, K.W. MORTON, AND E. SüLI, Euler characteristic Galerkin scheme with recovery, RAIRO Modél. Math. Anal. Numer., 27 (1993), pp. 863-894.
27. P. LIN, K.W. MORTON, AND E. SÜLI, Characteristic Galerkin schemes for scalar conservation laws in two and three space dimensions, SIAM J. Numer. Anal., 34 (1997), pp. 779-796.
28. T.A. MANTEUFFEL AND A.B. WHITE, JR., The numerical solution of second-order boundary value problems on nonuniform meshes, Math. Comp., 47 (1986), pp. 511-535.
29. K. MORGAN, J. PERAIRE, J. PEIRÖ, AND O. HASSAN, The computation of three dimensional flows using unstructured grids, Comput. Methods Appl. Mech. Engrg., 87 (1991), pp. 335- 352.
30. K.W. MORTON, Shock capturing, fitting and recovery, in Proceedings of the Eighth International Conference on Numerical Methods in Fluid Dynamics, Aachen, E. Krause, ed., Lecture Notes in Phys. 170, Springer-Verlag, New York, 1982, pp. 77-93.
31. K.W. MORTON, Characteristic Galerkin methods for hyperbolic problems, in Proceedings of Fifth GAMM Conference on Numerical Methods in Fluid Dynamics, Rome, M. Pandolfi and R. Piva, eds., Vieweg, Braunschwieg, Germany, 1983, pp. 243-250.
32. K.W. MORTON, Approximation of multidimensional hyperbolic partial differential equations, in Proceedings of the State of the Art in Numerical Analysis Conference, I.S. Duff and G.A. Watson, eds., Oxford University Press, Oxford, UK, 1996, pp. 473-502.
33. K.W. MORTON, Numerical Solution of Convection-Diffusion Problems, Appl. Math. Math. Comput. 12, Chapman & Hall, London, 1996.
34. K.W. MORTON AND P.N. CHILDS, Characteristic Galerkin methods for hyperbolic systems, in Nonlinear Hyperbolic Equations - Theory, Computation Methods, and Applications., J. Ballmann and R. Jeltsch, eds., Notes Numer. Fluid Mech. 24, Vieweg, Braunschwieg, Germany, 1989, pp. 435-455.
35. K.W. MORTON AND A.K. PARROTT, Generalised Galerkin methods for first-order hyperbolic equations, J. Comput. Phys., 36 (1980), pp. 249-270.
36. K.W. MORTON AND E. SÜLI, Evolution-Galerkin methods and their supraconvergence, Numer. Math., 71 (1995), pp. 331-355.
37. K.W. MORTON AND P.K. SWEBY, A comparison of flux limited difference methods and characteristic Galerkin methods for shock modelling, J. Comput. Phys., 73 (1987), pp. 203-230.
38. S. OSTKAMP, Multidimensional Characteristic Galerkin Schemes and Evolution Operators for Hyperbolic Systems, Ph.D. thesis, Universität Hannover, Hannover, Germany, 1995.
39. O. PIRONNEAU, On the transport-diffusion algorithm and its application to the Navier-Stokes equations, Numer. Math., 38 (1982), pp. 309-332.
40. V. SHANKAR, W. HALL, AND A.H. MOHAMMADIAN, A three-dimensionsal Maxwell's equation solver for computation of scattering from layered media, IEEE Trans, on Magnetics, 25 (1989), pp. 3098-3103.
41. A. TAFLOVE AND K.R. UMASHANKAR, The finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures, in Progress in Electromagnetics Research, Elsevier, Amsterdam, 1990, pp. 287-373.
42. B. VAN LEER, Towards the ultimate conservation difference scheme V. A second order sequel to Godunov's method, J. Comput. Phys., 23 (1979), pp. 101-136.
43. J.P. VILA, An analysis of a class of second-order accurate Godunov-type schemes, SIAM J. Numer. Anal., 26 (1989), pp. 830-853.
44. B. WENDROFF AND A.B. WHITE, JR., Some supraconvergent schemes for hyperbolic equations on irregular grids, in Second International Conference on Nonlinear Hyperbolic Problems, Aachen, J. Ballmann and R. Jeltsch, eds., Vieweg, Braunschwieg, Germany, 1989, pp. 671- 677.