A High-Order, Conservative Integrator with Local Time-Stepping

We present a family of multistep integrators based on the Adams--Bashforth methods. These schemes can be constructed for arbitrary convergence order with arbitrary step size variation. The step size can differ between different subdomains of the system. It can also change with time within a given subdomain. The methods are linearly conservative, preserving a wide class of analytically constant quantities to numerical roundoff, even when numerical truncation error is significantly higher. These methods are intended for use in solving conservative PDEs in discontinuous Galerkin formulations or in finite-difference methods with compact stencils. A numerical test demonstrates these properties and shows that significant speed improvements over the standard Adams--Bashforth schemes can be obtained.