Scalar auxiliary variable (SAV) stabilization of implicit-explicit (IMEX) time integration schemes for nonlinear structural dynamics

TL;DR

This paper introduces a scalar auxiliary variable (SAV) stabilization method for high-order IMEX schemes in nonlinear structural dynamics, achieving unconditional stability and high accuracy without iterative nonlinear solves.

Contribution

The paper proposes a novel SAV approach for high-order IMEX schemes that ensures unconditional stability and simplifies nonlinear problem solving in structural dynamics.

Findings

Achieves unconditional stability for nonlinear IMEX schemes.

Maintains high-order accuracy up to kth order.

Reduces computational cost by eliminating iterative nonlinear solves.

Abstract

Implicit-explicit (IMEX) time integration schemes are well suited for nonlinear structural dynamics because of their low computational cost and high accuracy. However, stability of IMEX schemes cannot be guaranteed for general nonlinear problems. In this article, we present a scalar auxiliary variable (SAV) stabilization of high-order IMEX time integration schemes that leads to unconditional stability. The proposed IMEX-BDFk-SAV schemes treat linear terms implicitly using kth-order backward difference formulas (BDFk) and nonlinear terms explicitly. This eliminates the need for iterations in nonlinear problems and leads to low computational cost. Truncation error analysis of the proposed IMEX-BDFk-SAV schemes confirms that up to kth-order accuracy can be achieved and this is verified through a series of convergence tests. Unlike existing SAV schemes for first-order ordinary differential equations (ODEs), we introduce a novel SAV for the proposed schemes that allows direct solution of the second-order ODEs without transforming them to a system of first-order ODEs. Finally, we demonstrate the performance of the proposed schemes by solving several nonlinear problems in structural dynamics and show that the proposed schemes can achieve high accuracy at a low computational cost while maintaining unconditional stability.