Multi-rate PDE methods for high Q oscillators

An integration formula for calculating the initial transient response of stiff systems exhibiting highly oscillatory solutions, such as quartz oscillators, is derived. The method reformulates the original system of ordinary differential-algebraic equations (DAEs) as a system of partial differential-algebraic equations (PDEs). The PDEs solve the original DAEs for a family of initial conditions. The novel algorithm has a rigorous mathematical basis. The time-scales of the solution of the PDEs, unlike the original DAEs, are not widely separated along the coordinate axes. The PDE approach is therefore much better suited for numerical simulation than the underlying DAEs. It can be considered as the inverse to the method of characteristics for PDEs. The straightforward extension to non-autonomous or driven autonomous cases is discussed briefly.