Abstract
We study a scalar hyperbolic partial differential equation, [Formula: see text], with nonlinear 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 nonlinear 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 nonlinear 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, althoug necessary for convergence at finite t, does 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.
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