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A scalar hyperbolic equation with GR-type non-linearity | | A.M. Khokhlov
; I.D. Novikov
; | | Date: |
18 Mar 2003 | | Journal: | Int.J.Mod.Phys. D12 (2003) 1889-1904 | | Subject: | gr-qc | | Affiliation: | Naval Research Laboratory, Washington DC, USA) and I.D. Novikov (Theoretical Astrophysics Center, Copenhagen, Denmark | | 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. | | Source: | arXiv, gr-qc/0303063 | | Services: | Forum | Review | PDF | Favorites | |
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