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

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 RungeKutta 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. Theoretical Astrophysics Center, Juliane Maries vej 30, DK-2100 Copenhagen, Denmark Copenhagen University Observatory, Juliane Maries vej 30, DK-2100 Copenhagen, Denmark Department of Astronomy and Astrophysics, The University of Chicago, 5640 Ellis Avenue, Chicago, IL 60637, USA NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, Denmark Astro Space Center of P.N. Lebedev Physical Institute, Profsoyouznaja 83/32, Moscow 118710, Russia