A local discontinuous Petrov-Galerkin method is proposed for solving three types of partial differential equations with second, third and fourth order derivatives, respectively. They are Burgers type equations, KdV type equations and bi-harmonic type equations. The method extends discontinuous Petrov-Galerkin method for conservation laws by rewriting corresponding equations into a first order system and solving the system instead of the original equation. The method has a fourth order accuracy and maintains advantages of discontinuous Petrov-Galerkin method. Numerical simulations verify that the method reaches optimal convergence order and simulates well complex wave interaction such as soliton propagation and collision.

A unified discontinuous Galerkin finite element framework on unstructured grids is developed to simulate gas-liquid twophase flows.In the framework,interior penalty discontinuous Galerkin (IPDG) method is employed to discretize imcompressible Navier-Stokes equations,while Runge-Kutta discontinuous Galerkin (RKDG) method is used to solve Level Set equation.Lid-driven cavity flow is simulated to validate IPDG method.Numerical results of bubble rising indicate the approach can be used to complex twophase flows with low computational efforts and simple implement.Moreover,RKDG method can effectively track deformations of interface without reinitialization.