A scalar hyperbolic equation with GR-type non-linearity

TL;DR

This paper investigates a scalar hyperbolic PDE with GR-like non-linearity, analyzing analytical solutions and proposing numerical schemes that improve stability and accuracy through variable transformation.

Contribution

It introduces two second-order accurate schemes for this PDE and demonstrates how exponential transformation enhances long-term stability and accuracy.

Findings

Numerical solutions converge at fixed times with increased resolution.

Asymptotic instability can occur despite second-order accuracy.

Exponential variable transformation improves stability and accuracy.

Abstract

We study a scalar hyperbolic partial differential equation with non-linear terms similar to those of the equations of general relativity. The equation has a number of non-trivial analytical solutions whose existence rely on a delicate balance between linear and non-linear terms. We formulate two classes of second-order accurate central-difference schemes, CFLN and MOL, for numerical integration of this equation. Solutions produced by the schemes converge to exact solutions at any fixed time $t$ when numerical resolution is increased. However, in certain cases integration becomes asymptotically unstable when $t$ is increased and resolution is kept fixed. This behavior is caused by subtle changes in the balance between linear and non-linear terms when the equation is discretized. Changes in the balance occur without violating second-order accuracy of discretization. We thus demonstrate that a second-order accuracy and convergence at finite $t$ do not guarantee a correct asymptotic behavior and long-term numerical stability. Accuracy and stability of integration are greatly improved by an exponential transformation of the unknown variable.