A new high order space derivative discretization for 3D quasi-linear hyperbolic partial differential equations

In this paper, we propose a new high accuracy numerical method of O(k2 + k2h2 + h4) for the solution of three dimensional quasi-linear hyperbolic partial differential equations, where k > 0 and h > 0 are mesh sizes in time and space directions respectively. We mainly discretize the space derivative terms using fourth order approximation and time derivative term using second order approximation. We describe the derivation procedure in details and also discuss how our formulation is able to handle the wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. The proposed method behaves like a fourth order method for a fixed value of (k/h2). Some examples and their numerical results are provided to justify the usefulness of the proposed method.