Finite difference method

Abstract: Natural convection flow of an absorbing fluid up a uniform porous medium supported by a semi-infinite, ideally transparent, vertical flat plate due to solar radiation is considered. Boundary-layer equations are derived using the usual Boussinesq approximation and accounting for applied incident radiation flux. A convection type boundary condition is used at the plate surface. These equations exhibit no similarity solution. However, the local similarity method is employed for the solution of the present problem so as to allow comparisons with previously published work. The resulting approximate nonlinear ordinary differential equations are solved numerically by a standard implicit iterative finite-difference method. Graphical results for the velocity and temperature fields as well as the boundary friction and Nusselt number are presented and discussed.

Abstract: This paper investigates the accurate numerical solution of the equations governing bed-load sediment transport. Two approaches: a steady and an unsteady approach are discussed and five different formulations within these frameworks are derived. A flux-limited version of Roe's scheme is used with the different formulations on a channel test problem and the results compared.

Abstract: Service life of concrete structures under chloride environment can be predicted by formulations based on the mechanism of chloride ion diffusion. This mechanism can be mathematically described using the partial differential equation (PDE) of the Fick’s second law. One-dimensional PDE can be solved analytically by assuming constant surface chloride ion concentration and constant diffusion coefficient. However, the solution becomes more complicated when two additional conditions are included, i.e., concrete cover repair or replacement and time dependent variation of the surface chloride ion concentration and diffusion coefficient. In this paper, a numerical finite difference based formulation is proposed to effectively accommodate these two additional conditions. By virtue of numerical computation, the nonlinear initial chloride ion concentration can be treated in point-wise manner and both the time dependent surface chloride ion concentration and diffusion coefficient can be iteratively updated. Based on a Crank–Nicolson scheme within the finite difference method, a proper formulation accounting for space-dependent diffusion coefficient was derived; chloride ion concentration profiles are obtained and the service life of repaired concrete structures under chloride environment is predicted. Numerical examples and observations are finally presented.

Abstract: The newly developed unifying discontinuous formulation named the correction pro- cedure via reconstruction (CPR) for conservation laws is extended to solve the Navier-Stokes equations for 3D mixed grids. In the current development, tetrahedrons and triangular prisms are considered. The CPR method can unify several popular high order methods including the dis- continuous Galerkin and the spectral volume methods into a more efficient differential form. By selecting the solution points to coincide with the flux points, solution reconstruction can be com- pletely avoided. Accuracy studies confirmed that the optimal order of accuracy can be achieved with the method. Several benchmark test cases are computed by solving the Euler and compress- ible Navier-Stokes equations to demonstrate its performance.