Numerical Implementation of the Multisymplectic Preissman Scheme and Its Equivalent Schemes

TL;DR

This paper investigates the multisymplectic Preissman scheme for the KdV equation, identifies convergence issues caused by potential functions, introduces an artificial boundary condition to improve practicality, and proposes new, more efficient schemes with better stability.

Contribution

It reveals the cause of convergence issues in the Preissman scheme, introduces an artificial boundary condition to enable practical implementation, and develops new schemes that are more efficient and stable.

Findings

Artificial boundary condition enables practical implementation.

New schemes are more efficient with less computational cost.

Explicit schemes exhibit strong numerical stability.

Abstract

We analyze the multisymplectic Preissman scheme for the KdV equation with the periodic boundary condition and show that the unconvergence of the widely-used iterative methods to solve the resulting nonlinear algebra system of the Preissman scheme is due to the introduced potential function. A artificial numerical condition is added to the periodic boundary condition. The added boundary condition makes the numerical implementation of the multisymplectic Preissman scheme practical and is proved not to change the numerical solutions of the KdV equation. Based on our analysis, we derive some new schemes which are not restricted by the artificial boundary condition and more efficient than the Preissman scheme because of less computing cost and less computer storages. By eliminating the auxiliary variables, we also derive two schemes for the KdV equation, one is a 12-point scheme and the other is an 8-point scheme. As the byproducts, we present two new explicit schemes which are not multisymplectic but still have remarkable numerical stable property.