To complement maturing technology for high-order spatial discretization for partial differential equations, new attention is turning to time integration. Recent progress in solvers can help us to realize the theoretical advantages of high-order and fully implicit schemes such as multi-stage Runge—Kutta or Galerkin-in-time. Building on such successes, both in solvers and high-level software, for first-order systems, we turn our attention to PDEs with second-order time derivatives.
Runge-Kutta-Nyström methods are a family of schemes for integrating second-order differential equations in time. Like Runge-Kutta methods, these methods comprise a broad family of schemes that include low and high order, implicit and explicit methods and methods with many different kinds of conservation or stability properties. Since they directly attack second derivatives rather than rewriting the problem as a first-order system with auxiliary variables, they lead to less work and lower memory usage.
We describe recent developments in Irksome, part of the Firedrake project, to formulate and automate the implementation of RKN methods and efficient solution processes. Examples will demonstrate the performance of these methods for some wave-like equations.