A characteristic finite element method (FEM) is proposed for nonlinear fractional convection-diffusion equations. With the characteristic technique and fractional FEM framework, a fully discrete characteristic finite element scheme is constructed. It is utilized to simulate physical systems and studied in contrast with conventional schemes. It is demonstrated that the method captures exact solutions of governing equations well. The method enjoys good stability and high accuracy even if convection dominates diffusion essentially.

An entropy stable scheme based on moving meshes is proposed for hyperbolic conservation laws. The method employs equidistribution principle to redistribute mesh points. Numerical solutions on new meshes are updated by using a conservative-interpolation formula. Entropy stable fluxes and third order strong stability-preserving Runge-Kutta time evolution method are employed to obtain numerical solutions at next time level. Several test problems are presented to demonstrate that the method not only improves resolution in discontinuous areas, but also reduces possible spurious oscillations.

A high resolution scheme is presented for shallow water equations. The scheme is based on entropy stable numerical flux with high order weighted essentially non-oscillatory (WENO) reconstruction at cell interfaces. A strong stability-preserving Runge-Kutta method is employed to advance in time. Several benchmark numerical examples demonstrate that the scheme is accurate and non-oscillatory.

Node placement method with bubble simulation can generate high-qualify node sets in complex domains.However,its efficiency needs to be increased.Several modifications were done to reduce the cost of simulation.Firstly,let viscosity coefficient c gradually increases instead of being taken as a constant.It speeds up convergency.Moreover,at the end of each round simulation,in which bubbles additions or deletions are operated,c is assigned to a small value in order to ensure quality of bubble distribution.Secondly,as solving ordinary differential equations that control movement of bubbles,a low order numerical algorithm is chosen.Finally,sort process of overlapping rate of bubbles is removed.It is replaced by setting only threshold for bubbles additions and deletions.Numerial examples show that computing cost decreases by approximately 40% and average quality of Delaunay triangulation corresponding to node set is over 0.9.It shows that the algorithms are efficient and generate node sets with high-quality.

For mesh generation of a two-parameter surface, anisotropic and non-uniform node placement method with bubble simulation is applied to optimize node distribution in parameter area. Then the parameter area is meshed with constrained Delaunay triangulation. Finally, according to the mapping method, two-parametric surface mesh is obtained. A second order Riemann metric tensor determines distribution of nodes in the parameter area. It could be co-generated with a three-dimensional surface metric tensor and gradient of surface functions. Numerical examples show that the node placement method with bubble simulation can generate node set meeting requirements of Riemann metric in parameter area. Nodes are meshed and mapped back into the surface. A high quality surface mesh is obtained.

In the light of molecular dynamics simulation and bubble meshing a node placement method, called node placement method with bubble simulation, is presented. In the method nodes are bubbles and bubbles are moved by interacting forces. With dynamic simulation centers of bubbles form a good-quality node set in the domain. And this process doesn't need updating mesh connection constantly. Examples show that uniform point sets and non-uniform point sets generated by this method have good construction and gradualness. And this method is adaptable to complex regions. These placed nodes can be used in meshless method. And triangular meshes can also be generated from these placed nodes in finite element method. Furthermore, for locality of interacting forces, this method is similar to short-range molecular dynamics simulation, and is easy to parallelization. Experimental result shows that the parallelization of this method is feasible.

We establish a practical mathematical model to compute optimal influence radius of weight functions in three dimensions. It is solved with linear basis and quadratic basis. Effects of error estimates, computing time and condition numbers of weight function with different influence radius on computation performance are investigated. Numerical examples demonstrate relability, efficiency and optimality of the influence radius.