Application of the relaxat ion met hod to model hydraulic jumps

A finite difference method, known as the relaxation method, is generalized from a class of relaxation schemes with the ultimate aim of numerically modelling hydraulic jumps at a phase interface. This method has been applied previously to model gravity currents arising from the instantaneous release of a dense volume of fluid. The relaxation scheme is an iterative, second order accurate, time-marching method which is able to capture shocks and interfaces without front tracking or calculation of the eigenvalues of the Jacobian matrix for the flux vector. In this paper, the relaxation scheme will be described, with specific attention paid to the new generalizations included to account for boundary conditions, spatially dependent flux terms, and simple forcing terms. Numerical results will be compared with simple theory using the inviscid Burgers equation, permitting the simplicity of the scheme to be portrayed through this example. The more general case of the shallow-water equations for a single layer in one spatial dimension will then be modelled numerically for an initial release boundary value problem to show that the method is widely applicable to problems involving two fluids of large density differences such as air and water where the systems are sufficiently decoupled.