Leo Hari Wiryanto

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 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.

Analytical solution of Boussinesq equations as a model of wave generation

When a uniform stream on an open channel is disturbed by existing of a bump at the bottom of the channel, the surface boundary forms waves growing splitting and propagating. The model of the wave generation can be a forced Korteweg de Vries (fKdV) equation or Boussinesq-type equations. In case the governing equations are approximated from steady problem, the fKdV equation is obtained. The model gives two solutions representing solitary-like wave, with different amplitude. However, phyically there is only one profile generated from that process. Which solution is occured, we confirm from unsteady model. The Boussinesq equations are proposed to determine the stabil solution of the fKdV equation. From the linear and steady model, its solution is developed to determine the analytical solution of the unsteady equations, so that it can explain the physical phenomena, i.e. the process of the wave generation, wave splitting and wave propagation. The solution can also determine the amplitude and wave speed of the ...

Jin-Xin relaxation method used to solve the one-dimensional inviscid Burgers equation

We consider the Burgers equation and seek for its numerical solutions. The relaxation method of Jin and Xin, called the Jin–Xin relaxation method, is tested to solve the Burgers equation. We find that the Jin–Xin relaxation method solves the Burgers equation successfully. In addition, the Jin–Xin relaxation method holds the analytical properties of the Burgers equation better than the standard Lax–Friedrichs finite-volume method does.

Steady flow over an arbitrary obstruction based on the gravity wave equations

We derive an analytical solution to a steady state problem of the gravity wave equations. An arbitrary bottom topography is considered. The problem is assumed to be one dimensional. The depth, discharge and topography elevation at the left-end of the space domain are assumed to have the same values as those at the right-end. We obtain that the fluid surface on the whole interior space domain remains horizontal and is not influenced by the topography shape when we use the gravity wave equations. Furthermore, the analytical solution that we derive is used to test the performance of a finite volume method. We find that the gravity wave equations give some advantages in comparison to the shallow water equations.