Unconditional error estimate of linearly-implicit and energy-preserving schemes for nonlocal wave equations

Abstract

Compared to the classical wave equation, the nonlocal wave equation incorporates a nonlocal operator and can capture a broader range of practical phenomena. However, this nonlocal formulation significantly increases the computational cost in numerical simulations, necessitating the development of efficient and accurate time integration schemes. Inspired by the newly developed generalized scalar auxiliary variable (GSAV) method in Refs. [8], [24] for dissipative systems, this paper uses the GSAV approach to construct linearly-implicit energy-preserving schemes for nonlocal wave systems. The developed numerical schemes only require solving linear equations with constant coefficients at each time step and are more efficient than the original SAV schemes [30] for wave equations. We also discuss the unique solvability, conduct a rigorous error analysis, and present numerical examples to demonstrate the accuracy, conservation and effectiveness of the obtained schemes.