Multirate timestepping methods for hyperbolic conservation laws

Journal of Scientific Computing

Volumn 33, Issue 3, 2007, Pages 239-278

  Constantinescu, Emil M ^a   Sandu, Adrian ^a  

^a State University   (United States)

Author keywords

Hyperbolic conservation laws; Multirate time integration; Nonlinear stability; Strong stability preservation

Indexed keywords

BURGERS EQUATIONS; HYPERBOLIC CONSERVATION LAWS; MULTIRATE TIME INTEGRATION; MULTIRATE TIMESTEPPING METHODS; NONLINEAR STABILITY; SPATIAL DOMAINS; STRONG STABILITY PRESERVATIONS;

ALGORITHMS; DISCRETE TIME CONTROL SYSTEMS; GRID COMPUTING; NUMBER THEORY; TIME DOMAIN ANALYSIS;

LINEAR STABILITY ANALYSIS;

EID: 35549008061     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-007-9151-y     Document Type: Article

References (54)

1. Andrus, J.: Numerical solution for ordinary differential equations separated into subsystems. SIAM J. Numer. Anal. 16, 605-611 (1979)
2. Andrus, J.: Stability of a multi-rate method for numerical integration of ODEs. Comput. Math. Appl. 25, 3-14 (1993)
3. Bartel, A., Günther, M.: A multirate W-method for electrical networks in state-space formulation. J. Comput. Appl. Math. 147, 411-425 (2002)
4. Berkvens, P., Botchev, M., Lioen, W., Verwer, J.: A zooming technique for wind transport of air pollution. Tech. report, Centrum voor Wiskundeen Informatica, 1999
5. Boris, J., Book, D.: Flux-corrected transport I. SHASTA, a fluid transport algorithm that works. J. Comput. Phys. 135, 172-186 (1997)
6. Chakravarthy, S., Osher, S.: Numerical experiments with the Osher upwind scheme for the Euler equations. AIAA J. 21, 241-1248 (1983)
7. Chakravarthy, S., Osher, S.: Computing with high-resolution upwind schemes for hyperbolic equations. Lect. Appl. Math. 22, 57-86 (1985)
8. Dawson, C., Kirby, R.: High resolution schemes for conservation laws with locally varying time steps. SIAM J. Sci. Comput. 22, 2256-2281 (2001)
9. Dormand, J., Prince, P.J.: A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6, 19-26 (1980)
10. Engquist, B., Tsai, R.: Heterogeneous multiscale methods for stiff ordinary differential equations. Math. Comput. 74, 1707-1742 (2005)
11. Engstler, C., Lubich, C.: Multirate extrapolation methods for differential equations with different time scales. Computing 58, 173-185 (1997)
12. Ferm, L., Lötstedt, P.: Space-time adaptive solution of first order PDES. J. Sci. Comput. 26, 83-110 (2006)
13. Ferracina, L., Spijker, M.: Stepsize restrictions for total-variation-boundedness in general Runge-Kutta procedures. Appl. Numer. Math. 53, 265-279 (2005)
14. Gear, C., Wells, D.: Multirate linear multistep methods. BIT 24, 484-502 (1984)
15. Godunov, S.: A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 271-290 (1959)
16. Goodman, J., LeVeque, R., Randall, J.: On the accuracy of stable schemes for 2D scalar conservation laws. Math. Comput. 45, 15-21 (1985)
17. Gottlieb, S., Shu, C.-W.: Total variation diminishing Runge-Kutta schemes. Math. Comput. 67, 73-85 (1998)
18. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89-112 (2001)
19. Günther, M., Hoschek, M.: ROW methods adapted to electric circuit simulation packages. In: ICCAM '96: Proceedings of the 7th International Congress on Computational and Applied Mathematics, pp. 159-170. Elsevier, Amsterdam (1997)
20. Günther, M., Kværnø, A., Rentrop, P.: Multirate partitioned Runge-Kutta methods. BIT 41, 504-514 (2001)
21. Günther, M., Rentrop, P.: Multirate ROW-methods and latency of electric circuits, Appl. Numer. Math. (1993)
22. Hairer, E.: Order conditions for numerical methods for partitioned ordinary differential equations. Numer. Math. 36, 431-445 (1981)
23. Hairer, E., Norsett, S., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems. Springer, Berlin (1993), Chapter II. 1
24. Hairer, E., Norsett, S., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Berlin (1993) Chapter. II. 15
25. Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357-393 (1983)
26. Higueras, I.: On strong stability preserving time discretization methods. J. Sci. Comput. 21, 193-223 (2004)
27. Hundsdorfer, W., Koren, B., van Loon, M.: A positive finite-difference advection scheme. J. Comput. Phys. 117, 35-46 (1995)
28. Hundsdorfer, W., Ruuth, S., Spiteri, R.: Monotonicity-preserving linear multistep methods. SIAM J. Numer. Anal. 41, 605-623 (2003)
29. Kato, T., Kataoka, T.: Circuit analysis by a new multirate method. Electr. Eng. Jpn. 126, 55-62 (1999)
30. Kirby, R.: On the convergence of high resolution methods with multiple time scales for hyperbolic conservation laws. Math. Comput. 72, 1239-1250 (2002)
31. Kværnø, A.: Stability of multirate Runge-Kutta schemes. Int. J. Differ. Equ. Appl. 1, 97-105 (2000)
32. Kværnø, A., Rentrop, P.: Low order multirate Runge-Kutta methods in electric circuit simulation (1999)
33. Laney, C.: Computational Gasdynamics. Cambridge University Press, Cambridge (1998)
34. LeVeque, R.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)
35. Logg, A.: Multi-adaptive Galerkin methods for ODEs I. SIAM J. Sci. Comput. 24, 1879-1902 (2003)
36. Logg, A.: Multi-adaptive Galerkin methods for ODEs II: Implementation and applications. SIAM J. Sci. Comput. 25, 1119-1141 (2003)
37. Logg, A.: Multi-adaptive Galerkin methods for ODEs III: A priory estimates. SIAM J. Numer. Anal. 43, 2624-2646 (2006)
38. Osher, A., Chakravarthy, S.: High resolution schemes and the entropy condition. SIAM J. Numer. Anal. 21, 955-984 (1984)
39. Osher, A., Chakravarthy, S.: Very high order accurate TVD schemes. In: Oscillation Theory, Computation, and Methods of Compensated Compactness. IMA Vol. Math. Appl., vol. 2, pp. 229-274. Springer, Berlin (1986)
40. Rice, J.: Split Runge-Kutta methods for simultaneous equations, J. Res. National Inst. Standards Technol. 60 (1960)
41. Roe, P.: Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43, 357-372 (1981)
42. Sand, J., Burrage, K.: A Jacobi waveform relaxation method for ODEs. SIAM J. Sci. Comput. 20, 534-552 (1998)
43. Savcenco, V., Hundsdorfer, W., Verwer, J.: A multirate time stepping strategy for parabolic PDE. Tech. Report MAS-E0516, Centrum voor Wiskundeen Informatica, 2005
44. Savcenco, V., Hundsdorfer, W., Verwer, J.: A multirate time stepping strategy for stiff ODEs. BIT (2006, to appear)
45. Shampine, L.: Conservation laws and the numerical solution of ODEs. Comput. Math. Appl. B 12, 1287-1296 (1986)
46. Shu, C.-W.: Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9, 1073-1084 (1988)
47. Shu, C.-W., Osher, S.: TVB uniformly high-order schemes for conservation laws. Math. Comput. 49, 105-121 (1987)
48. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439-471 (1988)
49. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes II. J. Comput. Phys. 83, 32-78 (1989)
50. Spiteri, R., Ruuth, S.: A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40, 469-491 (2002)
51. Sweby, P.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995-1011 (1984)
52. Tang, H.-Z., Warnecke, G.: A class of high resolution schemes for hyperbolic conservation laws and convection-diffusion equations with varying time and space grids. J. Comput. Math. 24, 121-140 (2006)
53. Vreugdenhil, C., Koren, B. (eds.): Numerical Methods for Advection-Diffusion Problems, vol. 45. Vieweg, Wiesbaden (1993)
54. Zalesak, S.: Fully multidimensional flux corrected transport algorithms for fluids. J. Comput. Phys. 31, 335-362 (1979)