Combining finite elements and geometric wave propagation in 1-D

Christiaan Stolk
University of Twente, Holland

Date: October 26, 2005

We consider the initial value problem for a strictly hyperbolic partial differential equation on the circle. We transform the equation to an operator valued ODE $du/dt = R(t)u$, where $R(t)$ is bounded. The transformation involves applying differential operators, solving an elliptic differential equation, and applying a coordinate transformation involving the characteristics, which can be done at cost $O(N)$. The resulting ODE is solved using a multiscale time-stepping method, which results in an algorithm with complexity $O(N)$ for the original hyperbolic equation.