A New High-Order Approximation for the Solution of Two-Space-Dimensional Quasilinear Hyperbolic Equations - R. K. Mohanty; Suruchi Singh (Advances in Mathematical Physics)
we propose a new high-order approximation for the solution of two-space-dimensional quasilinear hyperbolic partial differential equation of the form 𝑢𝑡𝑡=𝐴(𝑥,𝑦,𝑡,𝑢)𝑢𝑥𝑥+𝐵(𝑥,𝑦,𝑡,𝑢)𝑢𝑦𝑦+𝑔(𝑥,𝑦,𝑡,𝑢,𝑢𝑥,𝑢𝑦,𝑢𝑡), 0<𝑥, 𝑦<1, 𝑡>0 subject to appropriate initial and Dirichlet boundary conditions , where 𝑘>0 and ℎ>0 are mesh sizes in time and space directions, respectively. We use only five evaluations of the function 𝑔 as compared to seven evaluations of the same function discussed by (Mohanty et al., 1996 and 2001). 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. Some examples and their numerical results are provided to justify the usefulness of the proposed method.