Higher order compact implicit schemes for the wave equation

Abstract:

Higher order finite-difference techniques have associated large star systems which engender complications near the boundary. In the numerical solution of hyperbolic equations, such boundary conditions require careful treatment since errors or instabilities generated there will in general pollute the entire calculation. To circumvent this difficulty, we use a class of implicit schemes suggested by H.-O. Kreiss, which achieves the highest order of accuracy possible on the smallest (most compact) mesh system. Here we develop a scheme which approximates the wave equation, \[ {U_{tt}} = a(x,y,t){U_{xx}} + b(x,y,t){U_{yy}},\] with fourth order accuracy in space and time. After an appropriate factorization, the resulting set of equations are tridiagonal and hence easily solved. The tridiagonal nature also indicates that the boundary conditions do not create special difficulties. Numerical experiments demonstrate the expected order of convergence and fulfill our expectations on the treatment of boundary conditions. An experimental computation also demonstrates that our results hold on L-shaped domains.