A P-adaptive Discontinuous Galerkin Method Using Local Time-stepping Strategy Applied To The Shallow Water Equations

As a new finite element method to simulate hyperbolic conservation laws, discon-tinuous Galerkin method (DG). compared with the continuous galerkin method (CG), has a lot of potential advantages, such as local conservation, inherent parallelism and so on. However, it also has some disadvantages such as too many degrees of freedom (DOF). Too many degrees of freedom result in more computational burden. Especially higher order DG will lead to much more degrees of freedom which directly decrease the computational efficiency. Apart from the factor of DOF, the choice of time steps plays an important role in computational efficiency. The standard DG method uses the globally smallest time step and computational efficiency declines when those time steps satisfying local stability criterion in most areas are much bigger than the globally smallest one.The key to enhance the computational efficiency of DG is to solve the two con-straints mentioned above. In this paper, in order to enhance the computational efficien-cy of DG method applied to simulation of shallow water equations the following two research works have been carried out:(1) As for the time steps, local time-stepping strategy combined with a new single-step DG method based on space-time Taylor expansion is applied and at the same time the usage of combining slope limiter with this algorithm to prevent spurious oscilla-tions is presented which extends the range of application. Local time-stepping strategy features different time steps for different elements. Traditional multi-step Runge-Kutta DG method can also apply local time-stepping algorithm but to some extend the effi-ciency of this algorithm is limited because of some other limitations to the time steps. The single-step DG method based on space-time expansion can combine naturally with local time-stepping strategy. In this paper ID and2D shallow water equations are sim-ulated using this new DG method and local time-stepping strategy mentioned above and the effectivity of enhancing computational efficiency is verified. (2) As for DOF, one method of combining p-adaptive strategy with the new single-step DG method and local time-stepping strategy is presented. P-adaptation, which ad-justs adaptively the orders of approximation polynomials according to the space changes of numerical solutions, uses more DOF in some local areas where the changes of solu-tions are sharp and uses less DOF in most areas. P-adaptive strategy reduces the DOF on the whole and therefore enhances the computational efficiency. Furthermore, the application of combining dynamic p-adaptive strategy with local time-stepping DG a-gain greatly enhances the computational efficiency. In this paper two models of partial dam-break are simulated using p-adaptation and local time-stepping strategy and the results show on one hand the distributions of polynomials correctly correspond to the changes of numerical solutions and on the other hand the computational efficiency is again enhanced effectively.