A Staggered Discontinuous Galerkin Method with Local TV Regularization for the Burgers Equation

Abstract

The staggered discontinuous Galerkin (SDG) method has been recently developed for the numerical approximation of partial differential equations. An important advantage of such methodology is that the numerical solution automatically satisfies some conservation properties which are also satisfied by the exact solution. In this paper, we will consider the numerical approximation of the inviscid Burgers equation by the SDG method. For smooth solutions, we prove that our SDG method has the properties of mass and energy conservation. It is well-known that extra care has to be taken at locations of shocks and discontinuities. In this respect, we propose a local total variation (TV) regularization technique to suppress the oscillations in the numerical solution. This TV regularization is only performed locally where oscillation is detected, and is thus very efficient. Therefore, the resulting scheme will preserve the mass and energy away from the shocks and the numerical solution is regularized locally near shocks. Detailed description of the method and numerical results are presented.

High‐order numerical solution of viscous Burgers' equation using a Cole‐Hopf barycentric Gegenbauer integral pseudospectral method

“…Among the common numerical techniques used are finite difference schemes, differential quadrature methods, finite volume methods, finite element methods, Haar wavelet methods, spectral methods, pseudospectral methods, method of lines, meshless methods, Adomian decomposition methods, B-spline methods, discontinuous Galerkin methods, reproducing kernel functions, etc. 10,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] However, such methods either lack the exponential convergence enjoyed by spectral and pseudospectral methods or they enjoy exponential convergence rate in the spatial direction but suffer from low-order convergence rate in the temporal direction or suffer from degradation of the observed precision because of the ill conditioning of the employed numerical differential operators to the extent that the development of efficient preconditioners becomes extremely crucial or subject to serious time step restrictions that could be more severe than those predicted by the standard stability theory. [36][37][38][39][40][41][42][43][44][45][46][47][48] To avoid the ill conditioning of differential operators and the reduction in convergence rate for derivatives, an alternative direction to the aforementioned methods is to recast the partial differential equation into its integral formulation to take advantage of the well conditioning of integral operators and then discretize the latter using various discretization techniques.…”