Spatio-temporal scalar auxiliary variable approach for the nonlinear convection-diffusion equation with discontinuous Galerkin method

In this paper, a scalar auxiliary variable approach combining with a discontinuous Galerkin method is proposed to handle the gradient-type nonlinear term. The nonlinear convection-diffusion equation is used as the model. The proposed equivalent system can effectively handle the nonlinear convection term by incorporating the spatial and temporal information, globally. With the introduced auxiliary variable, the stability of the system can be simply characterized. In the space, according to the regularity of the system, an optimal accuracy is obtained with the discontinuous Galerkin method. Two different time discretization techniques, that is, backward Euler and linearly extrapolated Crank-Nicolson schemes, are separately considered with first order and second order accuracy. The proposed schemes are unconditionally stable with proper selected parameters. For the error estimates, the optimal convergence rates are rigorously proved. In the numerical experiments, the convergence information is confirmed and a benchmark problem with shock tendency is then followed with robustness demonstration.

Automated summary

In this paper, a scalar auxiliary variable approach combined with the discontinuous Galerkin method is proposed to handle the gradient-type nonlinear term in a nonlinear convection-diffusion equation. The proposed approach effectively incorporates spatial and temporal information to handle the nonlinear convection term and ensures system stability. The optimal accuracy is achieved with the discontinuous Galerkin method in space, and two different time discretization techniques are considered with first and second order accuracy. The proposed schemes are unconditionally stable, and optimal convergence rates are rigorously proved through error estimates. Numerical experiments confirm the convergence and demonstrate the robustness of the proposed approach in a benchmark problem with shock tendency.