A method of treating boundary singularities in time-dependent problems

A method is presented for treating singularities which occur in solutions of Parabolic partial differential equations due to sharp corners in the boundary. The method is essentially an extension of the method due to Motz (1946) for solving Elliptic problems and approximates to the analytical form of the singularity in terms of neighbouring function values at each time step. It is used in conjunction with the simple explicit finite-difference scheme and subsequently the overall method is explicit

On the fourth-order accurate compact ADI scheme for solving the unsteady Nonlinear Coupled Burgers' Equations

The two-dimensional unsteady coupled Burgers' equations with moderate to severe gradients, are solved numerically using higher-order accurate finite difference schemes; namely the fourth-order accurate compact ADI scheme, and the fourth-order accurate Du Fort Frankel scheme. The question of numerical stability and convergence are presented. Comparisons are made between the present schemes in terms of accuracy and computational efficiency for solving problems with severe internal and boundary gradients. The present study shows that the fourth-order compact ADI scheme is stable and efficient

On the Comparison between Compact Finite Difference and Pseudospectral Approaches for Solving Similarity Boundary Layer Problems

We introduce two methods based on higher order compact finite differences for solving boundary layer problems. The methods called compact finite difference relaxation method (CFD-RM) and compact finite difference quasilinearization method (CFD-QLM) are an alternative form of the spectral relaxation method (SRM) and spectral quasilinearization method (SQLM). The SRM and SQLM are Chebyshev pseudospectral-based methods which have been successfully used to solve boundary layer problems. The main objective of this paper is to give a comparison of the compact finite difference approach against the pseudo-spectral approach in solving similarity boundary layer problems. In particular, we seek to identify the most accurate and computationally efficient method for solving systems of boundary layer equations in fluid mechanics. The results of the two approaches are comparable in terms of accuracy for small systems of equations. For larger systems of equations, the proposed compact finite difference approaches are more accurate than the spectral-method-based approaches

Solving 2D Time-Fractional Diffusion Equations by Preconditioned Fractional EDG Method

Fractional differential equations play a significant role in science and technology given that several scientific problems in mathematics, physics, engineering and chemistry can be resolved using fractional partial differential equations in terms of space and/or time fractional derivative. Because of new developments in the analysis and understanding of many complex systems in engineering and sciences, it has been observed that several phenomena are more realistically and accurately described by differential equations of fractional order. Fast computational methods for solving fractional partial differential equations using finite difference schemes derived from skewed (rotated) difference operators have been extensively investigated over the years. The main aim of this paper is to examine a new fractional group iterative method which is called Preconditioned Fractional Explicit Decoupled Group (PFEDG) method in solving 2D time-fractional diffusion equations. Numerical experiments and comparison with other existing methods are given to confirm the superiority of our proposed method

Finding apparent horizons in numerical relativity

This paper presents a detailed discussion of the ``Newton's method'' algorithm for finding apparent horizons in 3+1 numerical relativity. We describe a method for computing the Jacobian matrix of the finite differenced H(h) function \H(\h) by symbolically differentiating the finite difference equations, giving the Jacobian elements directly in terms of the finite difference molecule coefficients used in computing \H(\h). Assuming the finite differencing scheme commutes with linearization, we show how the Jacobian elements may be computed by first linearizing the continuum H(h) equations, then finite differencing the linearized (continuum) equations. We find this symbolic differentiation method of computing the \H(\h) Jacobian to be {\em much\/} more efficient than the usual numerical-perturbation method, and also much easier to implement than is commonly thought. When solving the (discrete) \H(\h) = 0 equations, we find that Newton's method generally converges very rapidly, although there are difficulties when the initial guess contains high-spatial-frequency errors. Using 4th~order finite differencing, we find typical accuracies for the horizon position in the 10^{-5} range for \Delta \theta = \frac{\pi/2}{50}

High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics

The paper develops high-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamical (RHD) equations, built on the local Lax-Friedrich splitting, the WENO reconstruction, the physical-constraints-preserving flux limiter, and the high-order strong stability preserving time discretization. They are extensions of the positivity-preserving finite difference WENO schemes for the non-relativistic Euler equations. However, developing physical-constraints-preserving methods for the RHD system becomes much more difficult than the non-relativistic case because of the strongly coupling between the RHD equations, no explicit expressions of the primitive variables and the flux vectors, in terms of the conservative vector, and one more physical constraint for the fluid velocity in addition to the positivity of the rest-mass density and the pressure. The key is to prove the convexity and other properties of the admissible state set and discover a concave function with respect to the conservative vector replacing the pressure which is an important ingredient to enforce the positivity-preserving property for the non-relativistic case. Several one- and two-dimensional numerical examples are used to demonstrate accuracy, robustness, and effectiveness of the proposed physical-constraints-preserving schemes in solving RHD problems with large Lorentz factor, or strong discontinuities, or low rest-mass density or pressure etc.Comment: 39 pages, 13 figure

The new computer program for three dimensional relativistic hydrodynamical model

An effective computer program for three dimensional relativistic hydrodynamical model has been developed. It implements a new approach to the early hot phase of relativistic heavy-ion collisions. The computer program simulates time-space evolution of nuclear matter in terms of ideal-fluid dynamics. Equations of motions of hydrodynamics are solved making use of finite difference methods. Commonly-used algorithms of numerical relativistic hydrodynamics RHLLE and MUSTA-FORCE have been applied in simulations. To speed-up calculations, parallel processing has been made available for solving hydrodynamical equations. The test results of simulations for 3D, 2D and Bjorken expansion are reported in this paper. As a next step we plan to implement the hadronization algorithm by implementing the continuous particle emission for freeze-out and comparing it with Cooper-Frye formula.Comment: Quark Matter 2005 Poster Session Proceedin

Numerical Study of Three-dimensional Spatial Instability of a Supersonic Flat Plate Boundary Layer

The behavior of spatially growing three-dimensional waves in a supersonic boundary layer was studied numerically by solving the complete Navier-Stokes equations. Satisfactory comparison with linear parallel and non-parallel stability theories, and experiment are obtained when a small amplitude inflow disturbance is used. The three-dimensional unsteady Navier-Stokes equations are solved by a finite difference method which is fourth-order and second-order accurate in the convection and viscous terms respectively, and second-order accurate in time. Spanwise periodicity is assumed. The inflow disturbance is composed of eigenfunctions from linear stability theory. By increasing the amplitude of the inflow disturbance, nonlinear effects in the form of a relaxation type oscillation of the time signal of rho(u) are observed