Parallel Type Decomposition Scheme for Quasi-Linear Abstract Hyperbolic Equation

TL;DR

This paper introduces a parallel decomposition scheme for solving abstract hyperbolic equations with elliptic operators, enabling independent local solutions that are combined to approximate the global solution efficiently.

Contribution

It presents a novel parallel type decomposition scheme for hyperbolic equations with sum-operator elliptic parts, along with convergence proof and error estimates.

Findings

Scheme converges under natural conditions.

Error estimates for the approximate solution.

Parallel solution approach improves computational efficiency.

Abstract

Cauchy problem for an abstract hyperbolic equation with the Lipschitz continuous operator is considered in the Hilbert space. The operator corresponding to the elliptic part of the equation is a sum of operators $A_{1},\,A_{2},\,\ldots,\,A_{m}$. Each addend is a self-adjoint and positive definite operator. A parallel type decomposition scheme for an approximate solution of the stated problem is constructed. The main idea of the scheme is that on each local interval classic difference problems are solved in parallel (independently from each other) respectively with the operators $A_{1},\,A_{2},\,\ldots,\,A_{m}$. The weighted average of the received solutions is announced as an approximate solution at the right end of the local interval. Convergence of the proposed scheme is proved and the approximate solution error is estimated, as well as the error of the difference analogue for the first-order derivative for the case when the initial problem data satisfy the natural sufficient conditions for solution existence.