Relaxation method for unsteady convection-diffusion equations

We propose and implement a relaxation method for solving unsteady linear and nonlinear convection-diffusion equations with continuous or discontinuity-like initial conditions. The method transforms a convection-diffusion equation into a relaxation system, which contains a stiff source term. The resulting relaxation system is then solved by a third-order accurate implicitexplicit (IMEX) RungeKutta method in time and a fifth-order finite difference WENO scheme in space. Numerical results show that the method can be used to effectively solve convection-diffusion equations with both smooth structures and discontinuities.