Solution Of Hyperbolic PDEs Research Articles

The local meshless collocation method for numerical simulation of shallow water waves based on generalized equal width (GEW) equation

The current paper concerns to develop an efficient and robust numerical technique to solve the shallow water equation based on the generalized equal width (GEW) model. The considered model i.e. the generalized equal width (GEW) equation is a PDE that it can be classified in the category of hyperbolic PDEs. The solution of hyperbolic PDEs is similar to a fixed or moving wave. Thus, for solving these problems, a suitable numerical procedure that its basis functions are similar to a flat or shape wave should be selected. For this aim, the local collocation method via two different basis functions is utilized. First, the space derivative is approximated by the local collocation procedure that this manner yields a system of nonlinear ODEs depends on the time variable. Furthermore, the constructed system of ODEs is solved by a fourth-order algorithm to get high-numerical results. The mentioned process is applied on several test problems to verify the efficiency of the numerical formulation.

Variations on Hermite Methods for Wave Propagation

AbstractHermite methods, as introduced by Goodrich et al. in [15], combine Hermite interpolation and staggered (dual) grids to produce stable high order accurate schemes for the solution of hyperbolic PDEs. We introduce three variations of this Hermite method which do not involve time evolution on dual grids. Computational evidence is presented regarding stability, high order convergence, and dispersion/dissipation properties for each new method. Hermite methods may also be coupled to discontinuous Galerkin (DG) methods for additional geometric flexibility [4]. An example illustrates the simplification of this coupling for Hermite methods.

Shock dynamics in layered periodic media

Solutions of constant-coefficient nonlinear hyperbolic PDEs generically develop shocks, even if the initial data is smooth. Solutions of hyperbolic PDEs with variable coefficients can behave very differently. We investigate formation and stability of shock waves in a one-dimensional periodic layered medium by computational study of time-reversibility and entropy evolution. We find that periodic layered media tend to inhibit shock formation. For small initial conditions and large impedance variation, no shock formation is detected even after times much greater than the time of shock formation in a homogeneous medium. Furthermore, weak shocks are observed to be dynamically unstable in the sense that they do not lead to significant long-term entropy decay. We propose a characteristic condition for admissibility of shocks in heterogeneous media that generalizes the classical Lax entropy condition and accurately predicts the formation or absence of shocks in these media.

Solution of hyperbolic PDEs using a stable adaptive multiresolution method

An efficient adaptive multiresolution numerical method is described for solving systems of partial differential equations. The grid is dynamically adapted during the integration procedure so that only the relevant information is stored. The convection terms are discretised with high-resolution methods, thus ensuring boundedness. The proposed method is general, but is particularly useful for highly convective problems involving sharp moving fronts, a situation that frequently occurs in many chemical engineering problems, and where standard procedures may lead to unphysical oscillations in the computed solution. Numerical results for five test problems are presented to illustrate the efficiency and robustness of the method. The adaptive strategy is found to significantly reduce the computation time and memory requirements, as compared to the fixed grid approach.

An explicit, adaptive grid algorithm for one-dimensional initial value problems

A new explicit, adaptive grid algorithm for use in the solution of one-dimensional initial value problems is described. The method is based on the concept of tracking individual features of the physical solution. Properties of these real features are used to construct an artificial resolution function corresponding to a finite number of artificial features, each of which is then used to track a real feature. The artificial resolution function is constructed in such a way that it is analytically integrable, which allows a very stable and simple grid generation. Moving the artificial features is much more straightforward than trying to move the grid-points directly. In particular, features which interact, or even move off the grid can be treated adequately, without adversely affecting the monotonicity and smoothness of the grid. The results of several experiments are presented. These include a number of problems using test artificials, the aim of which is to mimic many of the situations involving sharp discontinuities which arise in the solution of hyperbolic PDEs. In addition, the algorithm is implemented in conjunction with a hydrodynamical test problem.