Comparison of hp-adaptive error estimates for second order hyperbolic systems

Abstract

Explicit methods are generally preferred for hyperbolic problems. With this in mind the finite element method of lumped masses instead of consistent masses has been implemented to discretize the wave equation. The diagonal mass matrix resulting from lumping leads to a special system of second-order ordinary differential equations. Instead of transforming this system to an equivalent first-order system as is typically done, we solve this directly by using an explicit Runge–Kutta–Nyström method that offers improved efficiency and less memory. Explicit a posteriori error estimators developed for elliptic and parabolic problems have been extended to the wave equation. They are seen to be asymptotically exact for the method of lumped masses for second-order hyperbolic problems. Computational results indicate that the estimates are asymptotically exact.