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

TL;DR

This paper develops high-order splitting methods for parabolic equations, overcoming previous barriers by constructing methods up to order 16 with complex coefficients, and demonstrating their superior accuracy and flexibility.

Contribution

It introduces new high-order splitting schemes beyond the traditional order 14 limit, including explicit construction of orders 6, 8, and 16, improving accuracy for parabolic PDEs.

Findings

Methods of order 14 are inherently bounded and sub-optimal.

Constructed explicit methods of order 6, 8, and 16 with improved accuracy.

Order 16 is achievable, surpassing the previous order barrier.

Abstract

We are concerned with the numerical solution obtained by splitting methods of 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 second-order barrier can be overcome by using splitting methods with complex-valued coefficients (with positive real parts). In this way, methods 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 and 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 methods with complex coefficients with positive real part by building explicitly a method of order 16 as a composition of methods of order 8.