Summation by parts methods for the spherical harmonic decomposition of the wave equation in arbitrary dimensions

Abstract

We investigate numerical methods for wave equations in $n+2$ spacetime dimensions, written in spherical coordinates, decomposed in spherical harmonics on $S^n$, and finite-differenced in the remaining coordinates $r$ and $t$. Such an approach is useful when the full physical problem has spherical symmetry, for perturbation theory about a spherical background, or in the presence of boundaries with spherical topology. The key numerical difficulty arises from lower-order $1/r$ terms at the origin $r=0$. As a toy model for this, we consider the flat space linear wave equation in the form $\dotπ=ψ'+pψ/r$, $\dotψ=π'$, where $p=2l+n$, and $l$ is the leading spherical harmonic index. We propose a class of summation by parts (SBP) finite differencing methods that conserve a discrete energy up to boundary terms, thus guaranteeing stability and convergence in the energy norm. We explicitly construct SBP schemes that are second and fourth-order accurate at interior points and the symmetry boundary $r=0$, and first and second-order accurate at the outer boundary $r=R$.

Figures (9)

• Figure 1: 2nd-order convergence in the energy norm of SBP2 for $p=6$, with the initial data given in the text. We show $|e_{\pi,2}(\cdot,t;h)|$ against $t$ at 5 different resolutions with $h$ decreasing by factors of 2 from $1/10$ to $1/160$. The 5 curves are on top of each other, demonstrating 2nd-order convergence. With the normalisation of Eq. (\ref{['Epi2L2def']}), they indicate the actual $L^2$ numerical error at resolution $h=1/10$ (meaning there are $\sim 40$ gridpoints across the wave packet). The equivalent curves for $\psi$ and for the Sarbach and Evans numerical methods are similar.
• Figure 2: 2nd-order pointwise convergence of the same evolution. We show $e_{\pi,2}(r,t;h)$ at $t=14.25$ against $r$. The 5 curves are on top of each other, demonstrating perfect 2nd-order convergence. They indicate the actual pointwise numerical error at resolution $h=1/10$. The equivalent curves for $\psi$ and for the Sarbach and Evans numerical methods are similar.
• Figure 3: 4th-order convergence in the energy norm of SBP41. We show $|e_{\pi,4}(\cdot,t;h)|$ against $t$, with all other details of the initial data and evolution as for the previous figure. The 5 curves are on top of each other, demonstrating perfect 4th-order convergence until $t\sim 12$, when the interaction of the tail of the Gaussian initial data with the outer boundary begins to dominate the error. The equivalent curve for $\psi$ looks similar, and the equivalent curves for SBP42 are identical until $t\sim 12$.
• Figure 4: 2nd-order convergence in the energy norm of SBP41 for $p=6$, after the wave has first interacted with the boundary. We show $|e_{\pi,2}(\cdot,t;h)|$ (instead of $e_4$) against $t$, with all other details as in the previous figure. The 5 curves are on top of each other, demonstrating approximate 2nd-order convergence after $t\sim 20$, when the error generated by the interaction of the tail of the Gaussian initial data with the outer boundary dominates the error. The equivalent curve for $\psi$ looks similar.
• Figure 5: 3rd-order convergence in the energy norm of SBP42 for $p=6$, after the wave has first interacted with the boundary. We show $|e_{\pi,3}(t;h)|$ against $h$, with all other details as in the previous figure. The 5 curves are on top of each other, demonstrating approximate 3rd-order convergence after $t\sim 20$.