Introduction

In this project, we consider conservation laws with source terms, often referred to as balance laws, a prototype of which is given by

\[q_t + f(q)_x = \psi(q,x)\]

where \(q \in \mathbb{R}^m\), \(f:\mathbb{R}^m \times \mathbb{R}^m \rightarrow \mathbb{R}^m\) and \(\psi(q,x)\) are the vector of the conserved quantities, the vector of the fluxes and the source term, respectively. Many numerical methods have been developed for the solution of the aforementioned system; some of the most successful are the high resolution Godunov-type methods based on the use of Riemann solvers and nonlinear limiters. These and other methods are generally based on flux-differencing and make explicit use of the flux function \(f\).

Recently, in the framework of the method of lines, Ketcheson et al. sharpclaw paper have proposed a new class of WENO high-order accurate propagation methods for the solution of systems with spatially varying flux:

\[\label{varflux} \kappa(x)q_t + f(q,x)_x = \psi(q,x),\]

and spatially varying linear systems not in conservation form:

\[\label{varlin} \kappa(x)q_t + A(x)q_x = \psi(q,x).\]

The wide applicability and advantageous properties of the method have been tested through numerical examples, including problems in nonconservative form, problems with spatially varying fluxes, and problems involving near-equilibrium solutions of balance laws, i.e. when the solution is close to a steady state and \(f(q,x)_x \approx \psi(q,x)\). In the latter case, if the scheme is well-balanced it preserves exactly (up to the truncation error) specific steady state solutions of the governing equations and calculates accurately small perturbations around these conditions.

In Ketcheson et al. sharpclaw paper, this property has been analyzed for the 2D shallow water equations with a bottom topography characterized by an isolated elliptical shaped hump. The well-balancing is achieved through the \(f\)-wave Riemann solver and a novel wave-slope WENO reconstruction procedure. However the proposed approach reduces the order of accuracy of the WENO discretization to second order. Therefore, for balance-laws close to steady state the scheme is well-balanced but it is just second-order accurate.

The aim of this project is to extend the well-balancing property to the fifth-order accurate WENO discretization for the 2D shallow water equations with slowly varying bottom topography. This achievement is a key feature for the applications of the new high-order accurate spatial discretizations to a wide range of geophysical flow problems, including tsunami modelling. Indeed, a tsunami is a large and often destructive ocean wave(s) resulting mainly from sudden vertical movement of the ocean floor. When an undersea earthquake or other major disturbance causes a section of ocean floor to rise or sink abruptly, the mass of water above the affected area is also suddenly displaced and large waves are formed. A wave is defined as a shallow water wave when the ratio between the water depth and its wave length gets very small. A tsunami can have very long wavelengths and because of this, it is a shallow-water wave.

NOTE: The new high-order accurate wave-propagation WENO schemes are part of the sharpclaw solver which has been incorporated in the PyClaw software.