A Frequency Accurate Finite Difference Scheme for Burgers Equation

A method is presented for designing a one-step, explicit finite difference scheme for solving the inviscid Burgers equation based on an a priori specification of dissipation and phase accuracy requirements. Frequency accurate temporal and spatial approximations with undetermined coefficients are used, together with a set of constraints that ensure that the approximations converge as the spatial and temporal grid sizes approach zero and satisfy the Lax Equivalence Theorem. A practical design of the difference scheme using a heuristic zero placement method, combined with a stability requirement, results in a linear matrix problem which is solved to obtain the undetermined coefficients. The partial differential equation itself provides the relationship between the temporal and spatial frequency dependence in the numerical approximation