Application of the Hybridized Discontinuous Galerkin Method for Solving One-Dimensional Coupled Burgers Equations

‎This paper is devoted to proposing hybridized discontinuous Galerkin (HDG) approximations for solving a system of coupled Burgers equations (CBE) in a closed interval‎. ‎The noncomplete discretized HDG method is designed for a nonlinear weak form of one-dimensional $x-$variable such that numerical fluxes are defined properly‎, ‎stabilization parameters are applied‎, ‎and broken Sobolev approximation spaces are exploited in this scheme‎. ‎Having necessary conditions on the stabilization parameters‎, ‎it is proven in a theorem and corollary that the proposed method is stable with imposed homogeneous Dirichlet and/or periodic boundary conditions to CBE‎. ‎The desired HDG method is stated by using the Crank-Nicolson method for time-variable discretization and the Newton-Raphson method for solving nonlinear systems‎. ‎Numerical experiences show that the optimal rate of convergence is gained for approximate solutions and their first derivatives‎.