Automatic Reformulation of ODEs to Systems of First-Order Equations

Abstract

Most numerical ODE solvers require problems to be written as systems of first-order differential equations. This normally requires the user to rewrite higher-order differential equations as coupled first-order systems. Here, we introduce the treeVar class, written in object-oriented Matlab, which is capable of algorithmically reformulating higher-order ODEs to equivalent systems of first-order equations. This allows users to specify problems using a more natural syntax and saves them from having to manually derive the first-order reformulation. The technique works by using operator overloading to build up syntax trees of expressions as mathematical programs are evaluated. It then applies a set of rules to the resulting trees to obtain the first-order reformulation, which is returned as another program. This technique has connections with algorithmic/automatic differentiation. We present how treeVar has been incorporated in Chebfun, greatly improving the ODE capabilities of the system.