A New Fast Numerical Method Based on Off-Step Discretization for Two-Dimensional Quasilinear Hyperbolic Partial Differential Equations

摘要

We report a new 3-level implicit compact numerical method of order four in time and four in space based on off-step discretization for the solution of two-space-dimensional quasilinear hyperbolic equation w(tt) = A(x, y, t, w) w(xx) + B(x, y, t, w) w(yy) + f(x, y, t, w(x), w(y), w(t)), A > 0, B > 0 defined in the region 0 < x, y < 1, t > 0. We require only 19 grid points for the unknown variable w(x, y, t) and two extra off-step points each in x-, y- and t-directions. The proposed method is directly applicable to two-dimensional hyperbolic equations with singular coefficients, which is the main attraction of our work. We do not require any fictitious points for computation. The proposed method when applied to a two-dimensional damped wave equation is shown to be unconditionally stable. Operator splitting method is used to solve damped wave equation. Many benchmark problems are solved to confirm the fourth-order convergence of the proposed method.