High-order accurate multi-sub-step implicit integration algorithms with dissipation control for second-order hyperbolic problems

TL;DR

This paper introduces high-order implicit multi-sub-step algorithms based on ESDIRK for second-order hyperbolic problems, achieving third-order per sub-step, dissipation control, and unconditional stability, with improved accuracy and efficiency.

Contribution

It develops and analyzes four new high-order implicit algorithms with s ≤ 6 sub-steps that avoid order reduction and do not require extra solutions for accelerations.

Findings

Algorithms achieve s-th order accuracy with s ≤ 6 sub-steps.

Proposed methods maintain stability and dissipation control.

Numerical examples confirm superior performance over existing methods.

Abstract

This paper proposes an implicit family of sub-step integration algorithms grounded in the explicit singly diagonally implicit Runge-Kutta (ESDIRK) method. The proposed methods achieve third-order consistency per sub-step and thus the trapezoidal rule is always employed in the first sub-step. This paper demonstrates for the first time that the proposed $ s $-sub-step implicit method with $ s\le6 $ can reach $ s $th-order accuracy when achieving dissipation control and unconditional stability simultaneously. Hence, this paper develops, analyzes, and compares four cost-optimal high-order implicit algorithms within the present $ s $-sub-step method using three, four, five, and six sub-steps. Each high-order implicit algorithm shares identical effective stiffness matrices to achieve optimal spectral properties. Unlike the published algorithms, the proposed high-order methods do not suffer from the order reduction for solving forced vibrations. Moreover, the novel methods overcome the defect that the authors' previous algorithms require an additional solution to obtain accurate accelerations. Linear and nonlinear examples are solved to confirm the numerical performance and superiority of four novel high-order algorithms.