A Local Laplace-Finite Difference Method for One-Dimensional Kelvin-Voigt Waves with Time-Dependent Viscosity: Stability and Convergence

Abstract

This Paper develops a local Laplace-finite difference method for a one-dimensional Kelvin-Voigt wave equation with time dependent viscosity. On each time slab, the viscosity is frozen at the endpoint average, which is a second-order approximation of the interval mean, and the resulting constant-coefficient local problem is advanced through a numerical inverse Laplace transform. Second-order centered finite differences are used in space. Both displacement and velocity are recovered from Laplace inversion; in particular, the velocity is inverted from  rather than reconstructed by a first-order time difference. For the homogeneous semi-discrete problem, an energy estimate shows that the exact frozen-coefficient propagator is dissipative for every time-step size, so it is free of the explicit CFL restriction. The effect of numerical Laplace inversion is treated separately as a quadrature perturbation. Under standard smoothness assumptions, endpoint averaging produces a local coefficient-freezing defect of order  and hence a global temporal error of order ; the centered spatial discretization contributes . Manufactured-solution tests recover spatial orders 2.001, 2.000, and 2.000 under successive mesh refinement and temporal orders 2.088, 2.018, and 2.004. A modified Talbot inversion with 24 nodes makes the inversion error negligible in the reported tests. For a transient forcing with , the proposed method with  and  agrees with a high-accuracy ODE reference to approximately  in time-history RMSE. These results support the method as an accurate CFL-free time propagator for smooth time-dependent Kelvin–Voigt media, while also identifying contour inversion and coefficient regularity as practical limitations.