Semi-implicit spectral deferred correction methods for ordinary differential equations

A semi-implicit formulation of the method of spectral deferred corrections (SISDC) for ordinary differential equations with both stiff and non-stiff terms is presented. Several modifications and variations to the original spectral deferred corrections method by Dutt, Greengard, and Rokhlin concerning the choice of integration points and the form of the correction iteration are presented. The stability and accuracy of the resulting ODE methods are explored analytically and numerically. The SISDC methods are intended to be combined with the method of lines approach to yield a flexible framework for creating higher-order semi-implicit methods for partial differential equations. A discussion and numerical examples of the SISDC method applied to advection-diffusion type equations are included. The results suggest that higher-order SISDC methods are more efficient than semi-implicit Runge-Kutta methods for moderately stiff problems in terms of accuracy per function evaluation.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  A. Prothero,et al.  On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations , 1974 .

[3]  B. Gustafsson.  The convergence rate for difference approximations to mixed initial boundary value problems , 1975 .

[4]  Randall J. LeVeque,et al.  Numerical methods based on additive splittings for hyperbolic partial differential equations , 1981 .

[5]  Robert D. Skeel,et al.  A Theoretical Framework for Proving Accuracy Results for Deferred Corrections , 1982 .

[6]  P. Woodward,et al.  The Piecewise Parabolic Method (PPM) for Gas Dynamical Simulations , 1984 .

[7]  J. M. Sanz-Serna,et al.  Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations , 1987 .

[8]  J. Butcher.  The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods , 1987 .

[9]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[10]  Ernst Hairer,et al.  Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations , 1988 .

[11]  S. Orszag,et al.  High-order splitting methods for the incompressible Navier-Stokes equations , 1991 .

[12]  Steven J. Ruuth,et al.  Implicit-Explicit Methods for Time-Dependent PDE''s , 1993 .

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

[14]  David I. Gottlieb,et al.  The Theoretical Accuracy of Runge-Kutta Time Discretizations for the Initial Boundary Value Problem: A Study of the Boundary Error , 1995, SIAM J. Sci. Comput..

[15]  Xiaolin Zhong,et al.  Semi-Implicit Runge-Kutta Schemes for non-autonomous differential equations in reactive flow computations , 1996 .

[16]  Willem Hundsdorfer,et al.  Stability of implicit-explicit linear multistep methods , 1997 .

[17]  David I. Gottlieb,et al.  On the Removal of Boundary Errors Caused by Runge-Kutta Integration of Nonlinear Partial Differential Equations , 1994, SIAM J. Sci. Comput..

[18]  Steven J. Ruuth,et al.  Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations , 1997 .

[19]  D. Pathria,et al.  The Correct Formulation of Intermediate Boundary Conditions for Runge-Kutta Time Integration of Initial Boundary Value Problems , 1997, SIAM J. Sci. Comput..

[20]  Uri M. Ascher,et al.  Computer methods for ordinary differential equations and differential-algebraic equations , 1998 .

[21]  Charalambos Makridakis,et al.  Implicit-explicit multistep methods for quasilinear parabolic equations , 1999, Numerische Mathematik.

[22]  G. Russo,et al.  Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations , 2000 .

[23]  L. Greengard,et al.  Spectral Deferred Correction Methods for Ordinary Differential Equations , 2000 .

[24]  M L Minion.  Higher-Order Semi-Implicit Projection Methods , 2001 .

[25]  M. Calvo,et al.  Linearly implicit Runge—Kutta methods for advection—reaction—diffusion equations , 2001 .

[26]  M. Minion,et al.  Accurate projection methods for the incompressible Navier—Stokes equations , 2001 .

[27]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[28]  L. Greengard,et al.  A new version of the fast multipole method for screened Coulomb interactions in three dimensions , 2002 .

[29]  Alexander Ostermann,et al.  Stability of W-methods with applications to operator splitting and to geometric theory , 2002 .

[30]  Mari Paz Calvo,et al.  Avoiding the order reduction of Runge-Kutta methods for linear initial boundary value problems , 2002, Math. Comput..

[31]  A. Bourlioux,et al.  High-order multi-implicit spectral deferred correction methods for problems of reactive flow , 2003 .

[32]  M. Carpenter,et al.  Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations , 2003 .

[33]  Michael L. Minion,et al.  Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics , 2004 .