High-order implicit weighted compact nonlinear scheme for nonlinear coupled viscous Burgers’ equations

Abstract

In this paper a high-order implicit weighted compact nonlinear scheme for nonlinear coupled viscous Burgers’ equations is presented. The fifth-order weighted compact nonlinear scheme is used for the spatial discretization, while the third-order diagonal implicit Runge–Kutta method is used for the time discretization. The generated nonlinear system is solved by the Jacobian-free Newton–Krylov nonlinear solver, which is composed of the outer Newton iteration method and the inner Krylov subspace iteration method. Stability analysis shows that the presented implicit weighted compact nonlinear scheme is unconditionally stable. Numerical results indicate that the implicit scheme can achieve the designed third-order accuracy in time and has a great advantage in the computation efficiency compared to the third-order explicit total variation diminishing Runge–Kutta weighted essentially non-oscillatory scheme. In addition, the implicit scheme can capture discontinuities and shock waves with high resolution and can solve Burgers’ equations with all kinds of Reynolds numbers.