This work investigates the response of two-dimensional, turbulent boundary layer characteristics over a flat plate to the presence of suspended particulate matter. Both phases are assumed to be interacting continua. That is, the carrier fluid equations are considered to be coupled with the particle-phase equations. A finite-difference technique with non-uniform grid has been employed for the so...

A hybrid finite difference algorithm for the Zakai equation is constructed to solve nonlinear filtering problems. The algorithm combines the splitting-up finite difference scheme and hierarchical sparse grid method to solve moderately high dimensional nonlinear filtering problems. When applying hierarchical sparse grid method to approximate bell-shaped solutions in most applications of nonlinea...

We present a code for solving the coupled Einstein-hydrodynamics equations to evolve relativistic, selfgravitating fluids. The Einstein field equations are solved in generalized harmonic coordinates on one grid using pseudospectral methods, while the fluids are evolved on another grid using shock-capturing finite difference or finite volume techniques. We show that the code accurately evolves e...

In this paper, we study a postprocessing procedure for improving accuracy of the finite volume element approximations of semilinear parabolic problems. The procedure amounts to solve a source problem on a coarser grid and then solve a linear elliptic problem on a finer grid after the time evolution is finished. We derive error estimates in the L and H norms for the standard finite volume elemen...

The OVERTURE Framework is an ob ject-oriented environment for solving PDEs on serial and parallel architectures. It is a collection of C++ libraries that enables the use of finite difference and finite volume methods at a level that hides the details of the associated data structures, as well its the details of the parallel implementation. It is based on the A++/P++ array class library and is d...

We derive explicit formulas for the eigenvalues and eigenvectors of the Discrete Laplacian on a rectangular grid for the standard finite difference and finite element methods in 1D, 2D, and 3D. Periodic, Dirichlet, Neumann, and mixed boundary conditions are all considered. We show how the higher dimensional operators can be written as sums of tensor products of one dimensional operators, and th...

The method of difference potentials was originally proposed by Ryaben’kii and can be interpreted as a generalized discrete version of the method of Calderon’s operators in the theory of partial differential equations. It has a number of important advantages; it easily handles curvilinear boundaries, variable coefficients, and non-standard boundary conditions while keeping the complexity at the ...

We consider the numerical approximation of singularly perturbed linear second order reaction-diffusion boundary value problems with a small shift(δ) in the undifferentiated term and the shift depends on the small parameter(ǫ). The presence of small parameter induces twin boundary layers. The problem is discretized using standard finite difference scheme on an uniform mesh and the retarded argum...