In this paper,we present the local discontinuous Galerkin method for solving Burgers＇ equation and the modified Burgers＇ equation.We describe the algorithm formulation and practical implementation of the local discontinuous Galerkin method in detail.The method is applied to the solution of the one-dimensional viscous Burgers＇ equation and two forms of the modified Burgers＇ equation.The numerical results indicate that the method is very accurate and efficient.
In this paper,a new discontinuous Galerkin method is developed for the parabolic equation with jump coefficients satisfying the continuous flow condition.Theoretical analysis shows that this method is L^(2) stable.When the finite element space consists of interpolative polynomials of degrees k,the convergent rate of the semi-discrete discontinuous Galerkin scheme has an order of δ(h^(k)).Numerical examples for both 1-dimensional and 2-dimensional problems demonstrate the validity of the new method.
The physical fields in porous materials under strong shock wave reaction are very complicated. We simulate such systems using the grain contact material point method. The complex temperature fields in the material are treated with the morphological characterization. To compare the structures and evolution of characteristic regimes under various temperature thresholds, we introduce two concepts, structure similarity and process similarity. It is found that the temperature pattern dynamics may show high similarity under various conditions. Within the same material, the structures and evolution of high-temperature regimes may show high similarity if the shock strength and temperature threshold are chosen appropriately. For process similarity in materials with high porosity, the required temperature threshold increases parabolically with the impact velocity. When the porosity becomes lower, the increasing rate becomes higher. For process similarity in different materials, the required temperature threshold and the porosity follow a power-law relationship in some range.
