On the energy estimates of semi-discrete wave equations with time dependent propagation speed

TL;DR

This paper investigates how time-dependent propagation speeds affect energy estimates in wave equations and their semi-discrete spatial discretizations, highlighting the impact of discretization on energy behavior.

Contribution

It provides new insights into the influence of time-dependent speeds on energy estimates in semi-discrete wave equations, bridging the gap between continuous and discrete models.

Findings

Energy estimates are significantly affected by time-dependent propagation speeds.

Discretization introduces additional effects on energy behavior compared to continuous models.

Results inform better numerical schemes for wave equations with variable speeds.

Abstract

Discretization is a fundamental step in numerical analysis for the problems described by differential equations, and the difference between the continuous model and discrete model is one of the most important problems. In this paper, we consider the difference in the effect of the time-dependent propagation speed on the energy estimate of the solutions for the wave equation and the semi-discrete wave equation which is a discretization with respect to space variables.