Multirate Explicit Adams Methods for Time Integration of Conservation Laws

This paper constructs multirate linear multistep time discretizations based on Adams-Bashforth methods. These methods are aimed at solving conservation laws and allow different timesteps to be used in different parts of the spatial domain. The proposed family of discretizations is second order accurate in time and has conservation and linear and nonlinear stability properties under local CFL conditions. Multirate timestepping avoids the necessity to take small global timesteps—restricted by the largest value of the Courant number on the grid—and therefore results in more efficient computations. Numerical results obtained for the advection and Burgers’ equations confirm the theoretical findings.

[1]  Andreas Bartel,et al.  A multirate W-method for electrical networks in state-space formulation , 2002 .

[2]  Eitan Tadmor,et al.  Strong Stability-Preserving High-Order Time Discretization , 2001 .

[3]  J. Dormand,et al.  A family of embedded Runge-Kutta formulae , 1980 .

[4]  John R. Rice,et al.  Split Runge-Kutta method for simultaneous equations , 1960 .

[5]  Willem Hundsdorfer,et al.  On monotonicity and boundedness properties of linear multistep methods , 2006, Math. Comput..

[6]  S. Osher,et al.  Very High Order Accurate TVD Schemes , 1986 .

[7]  R..  Split Runge-Kutta Method for Simultaneous Equations , 2010 .

[8]  S. Osher,et al.  Numerical experiments with the Osher upwind scheme for the Euler equations , 1982 .

[9]  R. Driscoll,et al.  Flowfield Experiments on a DF Chemical Laser , 1983 .

[10]  Toshiji Kato,et al.  Circuit analysis by a new multirate method , 1999 .

[11]  Willem Hundsdorfer,et al.  Monotonicity-Preserving Linear Multistep Methods , 2003, SIAM J. Numer. Anal..

[12]  Culbert B. Laney,et al.  Computational Gasdynamics: Waves , 1998 .

[13]  J. Kraaijevanger.  Contractivity of Runge-Kutta methods , 1991 .

[14]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[15]  D. Kinderlehrer,et al.  Oscillation theory, computation, and methods of compensated compactness , 1986 .

[16]  A. Harten.  High Resolution Schemes for Hyperbolic Conservation Laws , 2017 .

[17]  C. W. Gear,et al.  Multirate linear multistep methods , 1984 .

[18]  Roger Alexander,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems (E. Hairer, S. P. Norsett, and G. Wanner) , 1990, SIAM Rev..

[19]  S. Osher,et al.  Computing with high-resolution upwind schemes for hyperbolic equations , 1985 .

[20]  P. Rentrop,et al.  Multirate Partitioned Runge-Kutta Methods , 2001 .

[21]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[22]  Chi-Wang Shu.  TVB uniformly high-order schemes for conservation laws , 1987 .

[23]  J. F. Andrus,et al.  Numerical Solution of Systems of Ordinary Differential Equations Separated into Subsystems , 1979 .

[24]  P. Rentrop,et al.  Multirate ROW methods and latency of electric circuits , 1993 .

[25]  J. Verwer,et al.  A positive finite-difference advection scheme , 1995 .

[26]  H. Lenferink,et al.  Contractivity preserving explicit linear multistep methods , 1989 .

[27]  Emil M. Constantinescu,et al.  Multirate Timestepping Methods for Hyperbolic Conservation Laws , 2007, J. Sci. Comput..

[28]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[29]  Chi-Wang Shu.  TVB boundary treatment for numerical solutions of conservation laws , 1987 .

[30]  Kevin Burrage,et al.  A Jacobi Waveform Relaxation Method for ODEs , 1998, SIAM J. Sci. Comput..

[31]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[32]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[33]  H. Lenferink.  Contractivity-preserving implicit linear multistep methods , 1991 .

[34]  Sukumar Chakravarthy,et al.  High Resolution Schemes and the Entropy Condition , 1984 .

[35]  Christian Lubich,et al.  Multirate extrapolation methods for differential equations with different time scales , 1997, Computing.

[36]  Chi-Wang Shu.  Total-variation-diminishing time discretizations , 1988 .

[37]  J. F. Andrus.  Stability of a multi-rate method for numerical integration of ODE's , 1993 .

[38]  C. Vreugdenhil,et al.  Numerical methods for advection-diffusion problems , 1993 .