On the Exact Dimensional Splitting for a Scalar Quasilinear Hyperbolic Conservation Law

Abstract

A dimensional splitting scheme is applied to a multidimensional scalar homogeneous quasilinear hyperbolic equation (conservation law). It is analytically proved that the splitting error is zero. The proof is presented for the above partial differential equation in an arbitrary number of dimensions. A numerical example is given that illustrates the proved accuracy of the splitting scheme. In the example, the grid convergence of split (locally one-dimensional) compact and bicompact difference schemes and unsplit bicompact schemes combined with high-order accurate time stepping schemes (namely, Runge–Kutta methods of order 3, 4, and 5) is analyzed. The errors of the numerical solutions produced by these schemes are compared. It is shown that the orders of convergence of the split schemes remain high, which agrees with the conclusion that the splitting error is zero. Split compact and bicompact schemes are compared by their accuracy and computation speed.