Optimized high-order splitting methods for some classes of parabolic equations

We are concerned with the numerical solution obtained by splitting methodsof certain parabolic partial differential equations. Splitting schemes of order higher than two with real coefficients necessarily involve negative coefficients. It has been demonstrated that this se cond-order barrier can be overcome by using splitting methods with complex-valued coefficients (with positive real parts). In this way, method s of orders 3 to 14 by using the Suzuki‐Yoshida triple (and quadruple) jump composition procedure have been explicitly built. Here we reconsider this technique an d show that it is inherently bounded to order 14 and clearly sub-optimal with respect to error constants. As an alternative, we solve directly the algebraic equations arising from the order conditions and construct methods of orders 6 and 8 that are the most accurate ones available at present time, even when low accuracies are desired. We also show that, in the general case, 14 is not an order barrier for splitting method s with complex coefficients with positive real part by building explicitly a method of order 16 as a composition of methods of order 8.

[1]  Ander Murua,et al.  An algebraic theory of order , 2009 .

[2]  Fernando Casas,et al.  On the necessity of negative coefficients for operator splitting schemes of order higher than two , 2005 .

[3]  A. Ostermann,et al.  High order splitting methods for analytic semigroups exist , 2009 .

[4]  S. Fauve,et al.  Localized structures generated by subcritical instabilities , 1988 .

[5]  Q. Sheng.  Solving Linear Partial Differential Equations by Exponential Splitting , 1989 .

[6]  M. Suzuki,et al.  Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations , 1990 .

[7]  P. Matthews,et al.  Oscillatory pattern formation with a conserved quantity , 2005 .

[8]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[9]  Stéphane Descombes,et al.  Splitting methods with complex times for parabolic equations , 2009 .

[10]  Tao Tang,et al.  Error bounds for fractional step methods for conservation laws with source terms , 1995 .

[11]  C. Lubich,et al.  Error Bounds for Exponential Operator Splittings , 2000 .

[12]  John E. Chambers,et al.  Symplectic Integrators with Complex Time Steps , 2003 .

[13]  Christian Lubich,et al.  On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations , 2008, Math. Comput..

[14]  P. Palffy-Muhoray,et al.  The Complex Ginzburg-Landau Equation for Beginners , 1994 .

[15]  Stéphane Descombes,et al.  Strang's formula for holomorphic semi-groups , 2002 .

[16]  M. Suzuki,et al.  General theory of fractal path integrals with applications to many‐body theories and statistical physics , 1991 .

[17]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[18]  H. Holden,et al.  Splitting methods for partial differential equations with rough solutions : analysis and MATLAB programs , 2010 .

[19]  Fernando Casas,et al.  Splitting and composition methods in the numerical integration of differential equations , 2008, 0812.0377.

[20]  Tao Tang,et al.  Convergence Analysis for Operator-Splitting Methods Applied to Conservation Laws with Stiff Source Terms , 1998 .

[21]  Mechthild Thalhammer,et al.  High-Order Exponential Operator Splitting Methods for Time-Dependent Schrödinger Equations , 2008, SIAM J. Numer. Anal..

[22]  H. Yoshida.  Construction of higher order symplectic integrators , 1990 .

[23]  Creutz,et al.  Higher-order hybrid Monte Carlo algorithms. , 1989, Physical review letters.

[24]  J. M. Sanz-Serna,et al.  Order conditions for numerical integrators obtained by composing simpler integrators , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[25]  H. H. Rosenbrock,et al.  Some general implicit processes for the numerical solution of differential equations , 1963, Comput. J..

[26]  S. Blanes,et al.  Splitting methods with complex coefficients , 2010, 1001.1549.

[27]  Alexander Ostermann,et al.  Exponential splitting for unbounded operators , 2009, Math. Comput..

[28]  Tasso J. Kaper,et al.  N th-order operator splitting schemes and nonreversible systems , 1996 .

[29]  J. Verwer,et al.  Numerical solution of time-dependent advection-diffusion-reaction equations , 2003 .