Splitting Schemes for Some Second-Order Evolution Equations

TL;DR

This paper develops unconditionally stable splitting schemes for second-order evolution equations involving operator sums, enabling separate solutions for operators A and A* to improve computational stability and efficiency.

Contribution

It introduces a novel approach to constructing unconditionally stable schemes for second-order evolution equations with operator sums, using operator splitting and multiplicative perturbation techniques.

Findings

Schemes are unconditionally stable under general conditions.

Separate solutions for operators A and A* improve computational efficiency.

Application to elastic foundation problem demonstrates practical effectiveness.

Abstract

We consider the Cauchy problem for a second-order evolution equation, in which the problem operator is the sum of two self-adjoint operators. The main feature of the problem is that one of the operators is represented in the form of the product of operator A by its conjugate A*. Time approximations are carried out so that the transition to a new level in time was associated with a separate solution of problems for operators A and A*, not their products. The construction of unconditionally stable schemes is based on general results of the theory of stability (correctness) of operator-difference schemes in Hilbert spaces and is associated with the multiplicative perturbation of the problem operators, which lead to stable implicit schemes. As an example, the problem of the dynamics of a thin plate on an elastic foundation is considered.