Factorized schemes of second-order accuracy for numerical solving unsteady problems

TL;DR

This paper develops and analyzes second-order accurate numerical schemes for unsteady problems, focusing on stability, spectral properties, and computational efficiency, including modifications with first-degree operator polynomials.

Contribution

It introduces factorized, unconditionally stable schemes with spectral mimetic stability, including a two-stage predictor-corrector method using operator factorization.

Findings

The implicit Padé-based scheme is unconditionally stable and spectrally stable.

Modified schemes with first-degree operator polynomials are computationally advantageous.

Numerical experiments confirm the theoretical stability and accuracy of the proposed methods.

Abstract

Schemes with the second-order approximation in time are considered for numerical solving the Cauchy problem for an evolutionary equation of first order with a self-adjoint operator. The implicit two-level scheme based on the Pad\'{e} polynomial approximation is unconditionally stable. It demonstrates good asymptotic properties in time and provides an adequate evolution in time for individual harmonics of the solution (has spectral mimetic stability). In fact, the only drawback of this scheme is the necessity to solve an equation with an operator polynomial of second degree at each time level. We consider modifications of these schemes, which are based on solving equations with operator polynomials of first degree. Such computational implementations occur, for example, if we apply the fully implicit two-level scheme (the backward Euler scheme). A three-level modification of the SM-stable scheme is proposed. Its unconditional stability is established in the corresponding norms. The emphasis is on the scheme, where the numerical algorithm involves two stages, namely, the backward Euler scheme of first order at the first (prediction) stage and the following correction of the approximate solution using a factorized operator. The SM-stability is established for the proposed scheme. To illustrate the theoretical results of the work, a model problem is solved numerically.