Flux gradient terms and source terms are two fundamental components of hyperbolic systems of balance law. Though having distinct mathematical natures, they form and maintain an exact balance in a special class of solutions, which are called steady-state solutions. In this dissertation, we are interested in the construction of well-balanced schemes, which are the numerical methods for hyperbolic systems of balance laws that are capable of exactly preserving steady-state solutions on the discrete level. We first introduce a well-balanced scheme for the Euler equations of gas dynamics with gravitation. The well-balanced property of the designed scheme hinges on a reconstruction process applied to equilibrium variables---the quantities that stay constant at steady states. In addition, the amount of numerical viscosity is reduced in the areas where the flow is in (near) steady-state regime, so that the numerical solutions under consideration can be evolved in a well-balanced manner. We then consider the shallow water equations with friction terms, which become very stiff when the water height is close to zero. The stiffness in the friction terms introduces additional difficulty for designing an efficient well-balanced scheme. If treated explicitly, the stiff friction terms impose a severe restriction on the time step. On the other hand, a straightforward (semi-) implicit treatment of the stiff friction terms can greatly enhance the efficiency, but will break the well-balanced property of the resulting scheme. To this end, we develop a new semi-implicit Runge-Kutta time integration method that is capable of maintaining the well-balanced property under the time step restriction determined exclusively by non-stiff components in the underlying equations. The well-balanced property of our schemes are tested and verified by extensive numerical simulations, and notably, the obtained numerical results clearly indicate that the well-balanced property plays an important role in achieving high resolutions when a coarse grid is used. / acase@tulane.edu

acase@tulane.edu / Shallow water models are widely used to describe and study fluid dynamics phenomena where the horizontal length scale is much greater than the vertical length scale, for example, in the atmosphere and oceans. Since analytical solutions of the shallow water models are typically out of reach, development of accurate and efficient numerical methods is crucial to understand many mechanisms of atmospheric and oceanic phenomena. In this dissertation, we are interested in developing simple, accurate, efficient and robust numerical methods for two shallow water models --- the Saint-Venant system of shallow water equations and the two-mode shallow water equations. We first construct a new second-order moving-water equilibria preserving central-upwind scheme for the Saint-Venant system of shallow water equations. Special reconstruction procedure and source term discretization are the key components that guarantee the resulting scheme is capable of exactly preserving smooth moving-water steady-state solutions and a draining time-step technique ensures positivity of the water depth. Several numerical experiments are performed to verify the well-balanced and positivity preserving properties as well as the ability of the proposed scheme to accurately capture small perturbations of moving-water steady states. We also demonstrate the advantage and importance of utilizing the new method over its still-water equilibria preserving counterpart. We then develop and study numerical methods for the two-mode shallow water equations in a systematic way. Designing a reliable numerical method for this system is a challenging task due to its conditional hyperbolicity and the presence of nonconservative terms. We present several numerical approaches---two operator splitting methods (based on either Roe-type upwind or central-upwind scheme), a central-upwind scheme and a path-conservative central-upwind scheme---and test their performance in a number of numerical experiments. The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method for this system. / 1 / Yuanzhen Cheng