Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2003-03-18
Int.J.Mod.Phys. D12 (2003) 1889-1904
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
submitted to Class. Quantum Grav
Scientific paper
10.1142/S0218271803004390
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.
Khokhlov Alexei M.
Novikov Igor D.
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