Consistent Weighted Average Flux of Well-Balanced TVD-RK Discontinuous Galerkin Method for Shallow Water Flows

Thida Pongsanguansin,Montri Maleewong,Khamron Mekchay

doi:10.1155/2015/591282

Thida Pongsanguansin, Montri Maleewong + Show 1 more

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https://doi.org/10.1155/2015/591282

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Abstract

A well-balanced scheme with total variation diminishing Runge-Kutta discontinuous Galerkin (TVD-RK DG) method for solving shallow water equations is presented. Generally, the flux function at cell interface in the TVD-RK DG scheme is approximated by using the Harten-Lax-van Leer (HLL) method. Here, we apply the weighted average flux (WAF) which is higher order approximation instead of using the HLL in the TVD-RK DG method. The consistency property is shown. The modified well-balanced technique for flux gradient and source terms under the WAF approximations is developed. The accuracy of numerical solutions is demonstrated by simulating dam-break flows with the flat bottom. The steady solutions with shock can be captured correctly without spurious oscillations near the shock front. This presents the other flux approximations in the TVD-RK DG method for shallow water simulations.

Highlights

Hyperbolic balanced law for one-dimensional problem isUt + Fx (U) = G (U), (1)where U, F, and G represent solution vector, flux function, and source terms, respectively.In this work, the hyperbolic equation is the shallow water equations which can be expressed by ht + qx = 0, qt + ( q2 h gh2 2 ) x =
It has been shown in the previous section that the weighted average flux is consistent with the total variation diminishing (TVD) Runge Kutta (TVD-RK) DG method
We present the TVD-RK discontinuous Galerkin method (TVD-RK DG) for solving nonlinear shallow water equations

Summary

Introduction

Where U, F, and G represent solution vector, flux function, and source terms, respectively. The second-order Runge-Kutta discontinuous Gelerkin method with well-balanced scheme is proposed by Kesserwani and Liang [13] They presented the wetting and drying algorithms for solving one-dimensional problem. Xing and Shu [14] proposed a well-balanced finite volume method based on the weighted essentially nonoscillatory (WENO) scheme for solving one- and two-dimensional problems They applied the decomposing source term technique into the sum of several terms to ensure the numerical balance between flux difference and source terms. In order to solve the shallow water equations with source terms, we modified the well-balanced discontinuous Galerkin scheme proposed by [15] and related work by Xing and Shu [14].

Numerical Scheme for Shallow Water Equations without Source Term

Well-Balanced TVD-RK DG with WAF Scheme

Numerical Results

Conclusions