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Properties of four numerical schemes applied to a scalar nonlinear scalar wave equation with a GR-type nonlinearity

Jakob Hansen·Alexei Khokhlov·Igor Novikov
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Abstract

We study stability, dispersion and dissipation properties of four numerical schemes (Iterative Crank-Nicolson, 3'rd and 4'th order Runge-Kutta and Courant-Fredrichs-Levy Non-linear). By use of a Von Neumann analysis we study the schemes applied to a scalar linear wave equation as well as a scalar non-linear wave equation with a type of non-linearity present in GR-equations. Numerical testing is done to verify analytic results. We find that the method of lines (MOL) schemes are the most dispersive and dissipative schemes. The Courant-Fredrichs-Levy Non-linear (CFLN) scheme is most accurate and least dispersive and dissipative, but the absence of dissipation at Nyquist frequency, if fact, puts it at a disadvantage in numerical simulation. Overall, the 4'th order Runge-Kutta scheme, which has the least amount of dissipation among the MOL schemes, seems to be the most suitable compromise between the overall accuracy and damping at short wavelengths.

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Jakob Hansen, Alexei Khokhlov, Igor Novikov. 2005-03-04. Properties of four numerical schemes applied to a scalar nonlinear scalar wave equation with a GR-type nonlinearity. https://doi.org/10.1142/s021827180400502x

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