Computational Mathematics and Modeling, Vol. 29, No. 3, July, 2018
APPLICATION OF A QUASI-ACOUSTIC SCHEME TO SOLVE SHALLOW-WATER
EQUATIONS WITH AN UNEVEN BOTTOM
V. A. Isakov
We describe the application of an explicit homogeneous conservative quasi-acoustic scheme to numeri-
cal solution of one-dimensional shallow-water equations with an uneven bottom. The scheme performs
linear reconstruction of the numerical solution within a single numerical cell and partitions the linear re-
construction into small-perturbation horizontal layers. The quasi-acoustic scheme correctly reproduces
the physical solution in the neighborhood of the sonic point without invoking artificial regularizers or
tuning parameters. The scheme is verified on a number of test and prototype problems.
Keywords: system of equations of hyperbolic type, equations of shallow water over an uneven bottom,
numerical methods, quasi-acoustic scheme.
Shallow-water equations are a system of equations of hyperbolic type describing incompressible fluid flow
in regions with horizontal dimensions much greater than depth. These equations are widely used in flow calcu-
lations in rivers, lakes, seas, and tidal currents; in the modeling of the tsunami waves hitting the shoreline;
in weather forecasting calculations, and more.
The most common numerical methods for the solution of systems of equations are the finite-difference
methods , finite-volume methods [2–6], and finite-element methods [7–9]. Each of these methods has its
strengths and weaknesses and is accordingly applied in different domains.
The quasi-acoustic scheme can be classified as MUSCL — Monotonic Upstream-Centered Scheme for
Conservation Laws . Classical MUSCLs use piecewise-linear reconstruction of the numerical solution in-
side a single numerical cell. The flow across the cell boundary is typically calculated by solving the Riemann
problem. This approach produces a second-order scheme with oscillations possibly appearing in the neighbor-
hood of the discontinuous solutions. One of the ways to ensure monotonicity of the numerical solution is by
applying limiter functions that limit the slope angle of the linear functions within the numerical cell.
Unlike classical MUSCL, the quasi-acoustic scheme calculates the flow across the cell boundary by parti-
tioning the locally linear reconstruction of the solution into horizontal small-perturbation layers. Each partition
layer moves with its characteristic velocity over its background field following the solution of the linearized
equation system, which has been obtained without using limiter functions. The scheme has proved successful
for the numerical solution of the equations of fluid dynamics [11–13]. In this article, we consider the application
of the quasi-acoustic scheme to the numerical solution of shallow-water equations with an uneven bottom.
1. Statement of the Problem
The system of one-dimensional shallow-water equations with an uneven bottom represents the laws of mass
and momentum conservation in a rectangular region
Ω = X
× 0, T
. It has the form
Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia; e-mail: email@example.com.
Translated from Prikladnaya Matematika i Informatika, No. 56, 2017, pp. 72–89.
1046–283X/18/2903–0319 © 2018 Springer Science+Business Media, LLC 319