Mathematics – Numerical Analysis
Scientific paper
2010-10-12
Mathematics
Numerical Analysis
Scientific paper
Splitting a solution of the wave equation in $n+1$ space dimensions into spherical harmonic components on $n$-spheres and reducing to first order results in the pair of first-order partial differential equations in radius and time $\dot\pi=\psi'+p\psi/r$ and $\dot\psi=\pi'$ for each spherical harmonic, where $p=2l+n$, and $l$ is a spherical harmonic index. The $\psi/r$ term gives rise to numerical stability problems near the origin, and also poses the key numerical difficulty in related systems of equations. We propose a class of summation by parts finite differencing methods that converge pointwise, including at $r=0$, and are stable because they admit a discrete energy. We explicitly construct such schemes that are 2nd and 4th order accurate at interior points, and first and 2nd order accurate at the spherical outer boundary $r=R$. We use the projection method to impose a class of boundary conditions for which the discrete energy is non-increasing.
Garfinkle David
Gundlach Carsten
Martin-Garcia Jose M.
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