Sudi Mungkasi

Validation of ANUGA hydraulic model using exact solutions to shallow water wave problems

ANUGA is an open source and free software developed by the Australian National University (ANU) and Geoscience Australia (GA). This software is a hydraulic numerical model used to solve the two-dimensional shallow water equations. The numerical method underlying it is a finite volume method. This paper presents some validation results of ANUGA with respect to exact solutions to shallow water flow problems. We identify the strengths of ANUGA and comment on future work that may be taken into account for ANUGA development.

Analytical solutions involving shock waves for testing debris avalanche numerical models

Analytical solutions to debris avalanche problems involving shock waves are derived. The debris avalanche problems are described in two different coordinate systems, namely, the standard Cartesian and topography-linked coordinate systems. The analytical solutions can then be used to test debris avalanche numerical models. In this article, finite volume methods are applied as the numerical models. We compare the performance of the finite volume method with reconstruction of the conserved quantities based on stage, height, and velocity to that of the conserved quantities based on stage, height, and momentum for solving the debris avalanche problems involving shock waves. The numerical solutions agree with the analytical solution. In addition, both reconstructions lead to similar numerical results. This article is an extension of the work of Mangeney et al. (Pure Appl Geophys 157(6–8):1081–1096, 2000).

A study of well-balanced finite volume methods and refinement indicators for the shallow water equations

Water flows can be modelled mathematically and one available model is the shallow water equations. nThis thesis studies solutions to the shallow water equations analytically and numerically. The study is separated into three parts. n n nThe first part is about well-balanced finite volume methods to solve steady and unsteady state problems. A method is said to be well-balanced if it preserves an unperturbed steady state at the discrete level~cite{NPPN2006}. We implement hydrostatic reconstructions proposed by Audusse et al.~cite{ABBKP2004} for the well-balanced methods with respect to the steady state of a lake at rest. Four combinations of quantity reconstructions are tested. Our results indicate an appropriate combination of quantity reconstructions for dealing with steady and unsteady state problems~cite{MR2010}. n n nThe second part presents some new analytical solutions to debris avalanche problems~cite{MR2011DA,MR2012PAAG} and reviews the implicit Carrier--Greenspan periodic solution for flows on a sloping beach~cite{MR2012CG}. The analytical solutions to debris avalanche problems are derived using characteristics and a variable transformation technique. The analytical solutions are used as benchmarks to test the performance of numerical solutions. nFor the Carrier--Greenspan periodic solution, we show that the linear approximation of the Carrier--Greenspan periodic solution may result in large errors in some cases. If an explicit approximation of the Carrier--Greenspan periodic solution is needed, higher order approximations should be considered. We propose second order approximations of the Carrier--Greenspan periodic solution and present a way to get higher order approximations. n n nThe third part discusses refinement indicators used in adaptive finite volume methods to detect smooth and nonsmooth regions. In the adaptive finite volume methods, smooth regions are coarsened to reduce the computational costs and nonsmooth regions are refined to get more accurate solutions. We consider the numerical entropy production~cite{Puppo2004} and weak local residuals~cite{KKP2002} as refinement indicators. Regarding the numerical entropy production, our work is the first to implement the numerical entropy production as a refinement indicator into adaptive finite volume methods used to solve the shallow water equations. Regarding weak local residuals, we propose formulations to compute weak local residuals on nonuniform meshes. Our numerical experiments show that both the numerical entropy production and weak local residuals are successful as refinement indicators. n n nSome publications corresponding to this thesis are listed in the References~cite{MR2010,MR2011DA,MR2012PAAG,MR2012CG,MR2011NEP,MR2013NEP,MR2011IEEE}. n n 10.1017/S0004972713000750

Weak local residuals as smoothness indicators for the shallow water equations

Abstract The system of shallow water equations admits infinitely many conservation laws. This paper investigates weak local residuals as smoothness indicators of numerical solutions to the shallow water equations. To get a weak formulation, a test function and integration are introduced into the shallow water equations. We use a finite volume method to solve the shallow water equations numerically. Based on our numerical simulations, the weak local residual of a simple conservation law with a simple test function is identified to be the best as a smoothness indicator.

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.

On the relevance of a variational iteration method for solving the shallow water equations

We solve the shallow water equations using a variational iteration method due to Ji-Huan He. Advantages and disadvantages of the method are investigated. Some reference solutions are used to assess the performance of the variational iteration method. Variational iteration solutions are mesh-free, which makes the computation easy at any point at any time. However, we must understand the validity of these solutions for large time values. The goal of this paper is to verify the validity of variational iteration solutions, when the method is used to solve the shallow water equations.

A smoothness indicator for numerical solutions to the Ripa model

The Ripa model is the system of shallow water equations taking the water temperature fluctuations into account. For one-dimensional case, the Ripa model consists of three partial differential equations relating to three primitive variables, namely water depth, velocity, and temperature. The Ripa model is hyperbolic, and its solution can be discontinuous. When the Ripa model is solved using a conservative numerical method, the solution is usually diffusive around discontinuities. The diffusion at rough regions (around discontinuities, such as contact and shock discontinuities) makes the solution inaccurate. In practice, we want to know the places where the solution is accurate, and where it is inaccurate. That is, we want to know where the solution is smooth, and where it is rough. In this paper we propose the numerical entropy production to detect the smoothness of numerical solutions to the Ripa model. Numerical results show that the numerical entropy production is a robust smoothness indicator for numerical solutions to the Ripa model.

A modified Mohapatra-Chaudhry two-four finite difference scheme for the shallow water equations

A two-four finite difference scheme for Boussinesq equations was developed by Mohapatra and Chaudhry in 2004. This scheme is of course also applicable to solve the shallow water equations. However this scheme is not robust to deal with dry bed, that is, spurious oscillations appear around wet-dry areas. In this paper we propose a modified two-four finite difference scheme to solve the shallow water equations involving (almost) dry bed. The modified scheme has fewer number of divisions by zero or almost zero, and at the same time, only conserved quantities (mass and momentum) are used in the evolution of the new scheme. The modification lies on the discretisation of the momentum equation. We discretise the momentum equation using the momentum variable itself rather than using the velocity variable as done by Mohapatra and Chaudhry. Numerical results show that our proposed scheme is more robust for wetting and drying processes of the shallow water equations.

Adomian decomposition method used to solve the gravity wave equations

The gravity wave equations are considered. We solve these equations using the Adomian decomposition method. We obtain that the approximate Adomian decomposition solution to the gravity wave equations is accurate (physically correct) for early stages of fluid flows.

Numerical solution to the shallow water equations using explicit and implicit schemes

Shallow water flows can be modeled mathematically. One available model is the shallow water equations. Accurate solutions to the shallow water equations are desired. In this paper we solve the one-dimensional shallow water equations for flat topography. We present the performance of three numerical methods. The first is the implicit collocated finite difference method proposed by Crowhurst and Li (the Crowhurst-Li FDM). The second is the explicit collocated Lax-Friedrichs finite volume method (the Lax-Friedrichs FVM). The third is the explicit staggered finite volume method proposed by Stelling and Duinmeijer (the Stelling-Duinmeijer FVM). All three methods are stable. We find that the Lax-Friedrichs FVM is simplest in terms of computation. We also find that the Stelling-Duinmeijer FVM is more accurate in solving dam break problems than the other two methods.