UC Riverside

Hyperbolic balance laws are a special class of partial differential equations that represent various physical phenomena. Numerous scientific and engineering problems can be modeled using balance laws. This thesis contains three parts dedicated to the advancement of numerical methods for hyperbolic balance laws.

In the first and second parts, we construct high-order accurate well-balanced discontinuous Galerkin methods for two different systems of hyperbolic balance laws. The first part is devoted to the Ripa model and the second to the arterial blood flow model. The schemes were developed to preserve zero velocity and non-zero velocity steady states. The methods are an extension of the schemes used for the shallow water equations. Special attention is paid to the projection of the initial conditions into piecewise polynomial space, the approximation of the source term, and the construction of the numerical fluxes. High order and well-balanced methods have been previously developed for zero velocity steady states, however little work has been done on the more general non-zero velocity steady states. Numerical examples are given to demonstrate the well-balanced property, accuracy, non-oscillatory behavior at discontinuities, and ability to resolve small perturbations to steady states. This approach can be generalized to other balance laws.

In the third part, we develop an adjoint approach for recovering the topographical function included in the source term of one dimensional hyperbolic balance laws. We focus on a specific system, namely the shallow water equations, in an effort to recover the riverbed topography. The novelty of this work is the ability to robustly recover the bottom topography using only noisy boundary data from one measurement event and the inclusion of two regularization terms in the iterative update scheme. The adjoint scheme is determined from a linearization of the forward system and is used to compute the gradient of a cost function. The bottom topography function is recovered through an iterative process given by a three-operator splitting method which allows the feasibility to include two regularization terms. Numerous numerical tests demonstrate the robustness of the method regardless of the choice of initial guess and in the presence of discontinuities in the solution of the forward problem.