An adaptive viscosity regularization approach for the numerical solution of conservation laws: Application to finite element methods

We introduce an adaptive viscosity regularization approach for the numerical solution of systems of nonlinear conservation laws with shock waves. The approach seeks to solve a sequence of regularized problems consisting of the system of conservation laws and an additional Helmholtz equation for the artificial viscosity. We propose a homotopy continuation of the regularization parameters to minimize the amount of artificial viscosity subject to positivity-preserving and smoothness constraints on the numerical solution. The regularization methodology is combined with a mesh adaptation strategy that identifies the shock location and generates shock-aligned meshes, which allows to further reduce the amount of artificial dissipation and capture shocks with increased accuracy. We use the hybridizable discontinuous Galerkin method to numerically solve the regularized system of conservation laws and the continuous Galerkin method to solve the Helmholtz equation for the artificial viscosity. We show that the approach can produce approximate solutions that converge to the exact solution of the Burgers' equation. Finally, we demonstrate the performance of the method on inviscid transonic, supersonic, hypersonic flows in two dimensions. The approach is found to be accurate, robust and efficient, and yields very sharp yet smooth solutions in a few homotopy iterations.Comment: 42 pages, 22 figures, 4 table

High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs

Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching

On new and improved semi-numerical techniques for solving nonlinear fluid flow problems.

Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.Most real world phenomena is modeled by ordinary and/or partial differential equations. Most of these equations are highly nonlinear and exact solutions are not always possible. Exact solutions always give a good account of the physical nature of the phenomena modeled. However, existing analytical methods can only handle a limited range of these equations. Semi-numerical and numerical methods give approximate solutions where exact solutions are impossible to find. However, some common numerical methods give low accuracy and may lack stability. In general, the character and qualitative behaviour of the solutions may not always be fully revealed by numerical approximations, hence the need for improved semi-numerical methods that are accurate, computational efficient and robust. In this study we introduce innovative techniques for finding solutions of highly nonlinear coupled boundary value problems. These techniques aim to combine the strengths of both analytical and numerical methods to produce efficient hybrid algorithms. In this work, the homotopy analysis method is blended with spectral methods to improve its accuracy. Spectral methods are well known for their high levels of accuracy. The new spectral homotopy analysis method is further improved by using a more accurate initial approximation to accelerate convergence. Furthermore, a quasi-linearisation technique is introduced in which spectral methods are used to solve the linearised equations. The new techniques were used to solve mathematical models in fluid dynamics. The thesis comprises of an introductory Chapter that gives an overview of common numerical methods currently in use. In Chapter 2 we give an overview of the methods used in this work. The methods are used in Chapter 3 to solve the nonlinear equation governing two-dimensional squeezing flow of a viscous fluid between two approaching parallel plates and the steady laminar flow of a third grade fluid with heat transfer through a flat channel. In Chapter 4 the methods were used to find solutions of the laminar heat transfer problem in a rotating disk, the steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation and the classical von Kάrmάn equations for boundary layer flow induced by a rotating disk. In Chapter 5 solutions of steady two-dimensional flow of a viscous incompressible fluid in a rectangular domain bounded by two permeable surfaces and the MHD viscous flow problem due to a shrinking sheet with a chemical reaction, were solved using the new methods