Date of Award

Spring 2014

Document Type


Degree Name

Doctor of Philosophy (PhD)


Computational Analysis and Modeling

First Advisor

Don Liu


In this dissertation, two high order accurate numerical methods, Spectral Element Method (SEM) and Discontinuous Galerkin method (DG), are discussed and investigated. The advantages of both methods and their applicable areas are studied. Particular problems in complex geometry with complex physics are investigated and their high order accurate numerical solutions obtained by using either SEM or DG are presented. Furthermore, the Smoothed Particle Hydrodynamics (SPH) (a mesh-free weighted interpolation method) is implemented on graphics processing unit (GPU). Some numerical simulations of the complex flow with a free surface are presented and discussed to show the advantages of SPH method in handling rapid domain deformation.

In particular, four independent numerical examples are sequentially presented. A high-accurate SEM solution to the natural convection problem is provided. Up to the 6th order bases and the 4th order of the Runge-Kutta method are used in the simulation. Results show that our algorithm is more efficient than conventional methods, and the algorithm could obtain very detailed resolutions with moderate computional efforts (simply perform the hp-refinement). In another example, a more realistic and complete reaction model of simulating the reaction diffusion process in human neuromuscular junction (NMJ) is developed, and SEM is used to provide a high order accurate numerical solution for the model. Results have successfully predicted the distribution and amount of open receipts during a normal action potential, which helps us gain a better understanding of this process.

Still, high order DG method is used intensively to study the fluid problems with moderately high Reynolds (Re) number such as: flow passing a vertical cylinder and lid-driven cavity flow in both two dimensional (2D) and three dimensional (3D). Unstructured meshes (triangular element or tetrehedron) are adopted in our DG solver, which gives a greater ability than structured meshes (quadrilateral element or hexahedron) in solving particular problems with very complex geometry. By comparing our DG results with others obtained by conventional methods (Finite Difference Method, Finite Volume Method), high accuracy similar to other numerical results are obtained; however, the total number of degree of freedom in our simulation is greaterly reduced due to the spectral accuracy of the DG method.

Lastly, the SPH method is implemented on GPU to generate 2D and 3D simulations of fluid problems. The SPH solver has an advantage for solving fluid problems with complex geometries, rapid deformations and even discontinuities (wave-break) without generating computational grids. A noticeable speedup of our GPU implementation over the serial version on CPU is achieved. The solver is capable of developing further researches in real engineering problems such as: dam breaks, landslides, and near shore wave propagation and wave-structure interaction.