Boosting the accuracy of finite difference schemes via optimal time step selection and non-iterative defect correction

Abstract In this article, we present a unified analysis of the simple technique for boosting the order of accuracy of finite difference schemes for time dependent partial differential equations (PDEs) by optimally selecting the time step used to advance the numerical solution and adding defect correction terms in a non-iterative manner. The power of the technique, which is applicable to time dependent, semilinear, scalar PDEs where the leading-order spatial derivative has a constant coefficient, is its ability to increase the accuracy of formally low-order finite difference schemes without major modification to the basic numerical algorithm. Through straightforward numerical analysis arguments, we explain the origin of the boost in accuracy and estimate the computational cost of the resulting numerical method. We demonstrate the utility of optimal time step (OTS) selection combined with non-iterative defect correction (NIDC) on several different types of finite difference schemes for a wide array of classical linear and semilinear PDEs in one and more space dimensions on both regular and irregular domains.

[1]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[2]  Victor Pereyna.  Iterated deferred corrections for nonlinear boundary value problems , 1968 .

[3]  M. Doblaré,et al.  Fourth-order compact schemes with adaptive time step for monodomain reaction-diffusion equations , 2008 .

[4]  Murli M. Gupta,et al.  Symbolic derivation of finite difference approximations for the three‐dimensional Poisson equation , 1998 .

[5]  William E. Schiesser,et al.  Linear and nonlinear waves , 2009, Scholarpedia.

[6]  H. Kreiss,et al.  Time-Dependent Problems and Difference Methods , 1996 .

[7]  Bertil Gustafsson,et al.  Deferred Correction Methods for Initial Boundary Value Problems , 2002, J. Sci. Comput..

[8]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[9]  P. Hartman.  Ordinary Differential Equations , 1965 .

[10]  Kazufumi Ito,et al.  Higher-Order, Cartesian Grid Based Finite Difference Schemes for Elliptic Equations on Irregular Domains , 2005, SIAM J. Sci. Comput..

[11]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[12]  Guillermo Sapiro,et al.  Fourth order partial differential equations on general geometries , 2006, J. Comput. Phys..

[13]  S. Osher,et al.  A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) , 1999 .

[14]  Fuzhang Zhao.  Weighting factors for single-step trapezoidal method , 2006 .

[15]  David F. Griffiths,et al.  On the scope of the method of modified equations , 1986 .

[16]  A. Iserles.  A First Course in the Numerical Analysis of Differential Equations: Stiff equations , 2008 .

[17]  Wendy Kress.  Error estimates for deferred correction methods in time , 2007 .

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

[19]  R. Fedkiw,et al.  A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem , 2005 .

[20]  Graham F. Carey,et al.  Extension of high‐order compact schemes to time‐dependent problems , 2001 .

[21]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[22]  Lawrence F. Shampine,et al.  Error Estimation and Control for ODEs , 2005, J. Sci. Comput..

[23]  Mahidhar Tatineni,et al.  Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier-Stokes equations , 2007, J. Comput. Phys..

[24]  Bertil Gustafsson,et al.  Deferred Correction in Space and Time , 2002, J. Sci. Comput..

[25]  J. Cash,et al.  The MOL solution of time dependent partial differential equations , 1996 .

[26]  L. Chambers.  Linear and Nonlinear Waves , 2000, The Mathematical Gazette.

[27]  Xiaolin Zhong,et al.  Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation , 2005 .

[28]  Mikko Karttunen,et al.  Stencils with isotropic discretization error for differential operators , 2006 .

[29]  Per Lötstedt,et al.  High order accurate solution of the incompressible Navier-Stokes equations , 2005 .

[30]  G. Psihoyios,et al.  Solving time dependent PDEs via an improved modified extended BDF scheme , 2007, Appl. Math. Comput..

[31]  H. Stetter.  The defect correction principle and discretization methods , 1978 .

[32]  Stability analysis of finite difference schemes for the advection-diffusion equation , 1983 .

[33]  L. Falkenstein,et al.  Higher-Order Logic , 2021, Logic Works.

[34]  Heinz-Otto Kreiss,et al.  Difference Approximations for the Second Order Wave Equation , 2002, SIAM J. Numer. Anal..