Faster SDC convergence on non-equidistant grids by DIRK sweeps

Spectral deferred correction methods for solving stiff ODEs are known to converge reasonably fast towards the collocation limit solution on equidistant grids, but show a less favourable contraction on non-equidistant grids such as Radau-IIa points. We interprete SDC methods as fixed point iterations for the collocation system and propose new DIRK-type sweeps for stiff problems based on purely linear algebraic considerations. Good convergence is recovered also on non-equidistant grids. The properties of different variants are explored on a couple of numerical examples.

[1]  K. H. Schild,et al.  Gaussian collocation via defect correction , 1990 .

[2]  J. S. Wang.  Statistical Theory of Superlattices with Long-Range Interaction. I. General Theory , 1938 .

[3]  P. Zadunaisky.  On the estimation of errors propagated in the numerical integration of ordinary differential equations , 1976 .

[4]  W. Auzinger,et al.  Superconvergent Defect Correction Algorithms , 2004 .

[5]  Matthew M. Bowen,et al.  A Spectral Deferred Correction Method for Solving Cardiac Models , 2011 .

[6]  Christoph W. Ueberhuber,et al.  Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations , 1977 .

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

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

[9]  Anders C. Hansen,et al.  On the order of deferred correction , 2011 .

[10]  Colin B. Macdonald,et al.  Parallel High-Order Integrators , 2010, SIAM J. Sci. Comput..

[11]  Jingfang Huang,et al.  Arbitrary order Krylov deferred correction methods for differential algebraic equations , 2007, J. Comput. Phys..

[12]  Benjamin W. Ong,et al.  COMMENTS ON HIGH-ORDER INTEGRATORS EMBEDDED WITHIN INTEGRAL DEFERRED CORRECTION METHODS , 2009 .

[13]  Tao Tang,et al.  High-Order Convergence of Spectral Deferred Correction Methods on General Quadrature Nodes , 2013, J. Sci. Comput..

[14]  Peter Deuflhard,et al.  Adaptive Numerical Solution of PDEs , 2012 .

[15]  P. Rentrop,et al.  Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations , 1979 .

[16]  Michael L. Minion,et al.  TOWARD AN EFFICIENT PARALLEL IN TIME METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS , 2012 .

[17]  Jingfang Huang,et al.  Accelerating the convergence of spectral deferred correction methods , 2006, J. Comput. Phys..

[18]  Michael L. Minion,et al.  Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations , 2005 .

[19]  Winfried Auzinger,et al.  Modified defect correction algorithms for ODEs. Part II: Stiff initial value problems , 2005, Numerical Algorithms.

[20]  Alexander Ostermann,et al.  Rosenbrock methods for partial differential equations and fractional orders of convergence , 1993 .

[21]  Bertil Gustafsson,et al.  Deferred Correction Methods for Initial Value Problems , 2001 .

[22]  Jingfang Huang,et al.  Semi-implicit Krylov deferred correction methods for differential algebraic equations , 2012, Math. Comput..

[23]  Michael L. Minion,et al.  Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods , 2007 .

[24]  Winfried Auzinger,et al.  Modified Defect Correction Algorithms for ODEs. Part I: General Theory , 2004, Numerical Algorithms.