Numerical entropy production as smoothness indicator for shallow water equations

The numerical entropy production (NEP) for shallow water equations (SWE) is discussed and implemented as a smoothness indicator. We consider SWE in three different dimensions, namely, one-dimensional, one-and-a-half-dimensional, and two-dimensional SWE. An existing numerical entropy scheme is reviewed and an alternative scheme is provided. We prove the properties of these two numerical entropy schemes relating to the entropy steady state and consistency with the entropy equality on smooth regions. Simulation results show that both schemes produce NEP with the same behaviour for detecting discontinuities of solutions and perform similarly as smoothness indicators. An implementation of the NEP for an adaptive numerical method is also demonstrated. doi:10.1017/S1446181119000154

NUMERICAL ENTROPY PRODUCTION AS SMOOTHNESS INDICATOR FOR SHALLOW WATER EQUATIONS

The numerical entropy production (NEP) for shallow water equations (SWE) is discussed and implemented as a smoothness indicator. We consider SWE in three different dimensions, namely, one-dimensional, one-and-a-half-dimensional, and two-dimensional SWE. An existing numerical entropy scheme is reviewed and an alternative scheme is provided. We prove the properties of these two numerical entropy schemes relating to the entropy steady state and consistency with the entropy equality on smooth regions. Simulation results show that both schemes produce NEP with the same behaviour for detecting discontinuities of solutions and perform similarly as smoothness indicators. An implementation of the NEP for an adaptive numerical method is also demonstrated.

Adaptive Finite Volume Method for the Shallow Water Equations on Triangular Grids

This paper presents a numerical entropy production (NEP) scheme for two-dimensional shallow water equations on unstructured triangular grids. We implement NEP as the error indicator for adaptive mesh refinement or coarsening in solving the shallow water equations using a finite volume method. Numerical simulations show that NEP is successful to be a refinement/coarsening indicator in the adaptive mesh finite volume method, as the method refines the mesh or grids around nonsmooth regions and coarsens them around smooth regions.

An Accurate Smoothness Indicator for Shallow Water Flows along Channels with Varying Width

We extend the application of numerical entropy production, as a smoothness indicator, from conservation laws to balance laws. We aim to indicate the smoothness of solutions to the shallow water equations involving varying width, which are a system of balance laws. The numerical entropy production appears to be accurate to detect discontinuities. As a numerical test, a radial dam break is considered. We assume that there is a higher level of water inside a radial dam than water outside the dam wall. If the radial dam is totally broken, then water flows from inside to outside. The flow results in a solution having shock discontinuities. Finding the positions of the discontinuities is our interest. They are the positions where numerical solutions, such as those generated by a finite volume method, decrease their accuracy. Detecting the position of the discontinuity can help in the improvement of the numerical solution in terms of its accuracy. We obtain that the numerical entropy production is simple to implement but give an accurate detection. The discontinuity of the stage (free water surface) is clearly detected by large values of the numerical entropy production as the smoothness indicator.

Numerical Entropy and Adaptivity for Finite Volume Schemes

We propose an a-posteriori error/smoothness indicator for standard semi-discrete finite volume schemes for systems of conservation laws, based on the numerical production of entropy. This idea extends previous work by the first author limited to central finite volume schemes on staggered grids. We prove that the indicator converges to zero with the same rate of the error of the underlying numerical scheme on smooth flows under grid refinement. We construct and test an adaptive scheme for systems of equations in which the mesh is driven by the entropy indicator. The adaptive scheme uses a single nonuniform grid with a variable timestep. We show how to implement a second order scheme on such a space-time non uniform grid, preserving accuracy and conservation properties. We also give an example of a p-adaptive strategy.

A New Numerical Scheme for Linear Scalar Conservation Laws

In this paper, we develop a numerical scheme with the fifth-order polynomial reconstruction satisfying two conservation laws for the linear advection equation. The scheme is the Godunov type, and has two numerical entities, numerical solution and numerical entropy. Numerical experiments show that the scheme is more robust in long-time behaviors than that of [8].