Dissipation-Preserving Rational Spectral-Galerkin Method for Strongly Damped Nonlinear Wave System Involving Mixed Fractional Laplacians in Unbounded Domains

摘要

Abstract This paper aims at developing a dissipation-preserving, linearized, and time-stepping-varying spectral method for strongly damped nonlinear wave system in multidimensional unbounded domains Rddocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathbb {R}}^d$$end{document} (d=1, 2, and 3), where the nonlocal nature is described by the mixed fractional Laplacians. Because the underlying solutions of the problem involving mixed fractional Laplacians decay slowly with certain power law at infinity, we employ the rational spectral-Galerkin method using rational basis (or mapped Gegenbauer functions) for the spatial approximation. To capture the intrinsic dissipative properties of the model equations, we combine the Crank-Nicolson scheme with exponential scalar auxiliary variable approach for the temporal discretization. Based on the rate of nonlocal energy dissipation, we design a novel time-stepping-varying strategy to enhance the efficiency of the scheme. We present the detailed implementation of the scheme, where the main building block of the stiffness matrices is based on the Laguerre-Gauss quadrature rule for the modified Bessel functions of the second kind. The existence, uniqueness, and nonlocal energy dissipation law of the fully discrete scheme are rigourously established. Numerical examples in 3D case are carried out to demonstrate the accuracy and efficiency of the scheme. Finally, we simulate the nonlinear behaviors of 2D/3D dissipative vector solitary waves for damped sine-Gordon system I, for damped sine-Gordon system II, and for damped Klein–Gordon system to provide a deeper understanding of nonlocal physics.