Geometry of Numerical Complex Time Integration
This work provides a new geometric perspective on numerical integration methods, potentially improving efficiency for ODE solvers, but the results are theoretical and incremental.
The paper derives special complex time grids for Runge-Kutta methods that achieve superconvergence for linear autonomous ODEs, and extends the approach to arbitrary ODEs, linking it to composition methods with complex coefficients.
We are studying Runge-Kutta methods along complex paths of integration from a geometric point of view. Thereby we derive special complex time grids, which applied to the problem of integrating a linear autonomous system of ordinary differential equations, can be used to achieve a classical superconvergence effect. The approach is also adapted for arbitrary ODEs. Furthermore we draw a connection from our geometric reasoning to the class of composition methods with complex coefficients. Thereby, our main goal is to introduce a new point of view on these methods.