Finite Difference Scheme for Parabolic Problems on Composite Grids with Refinement in Time and Space (original) (raw)
Numerical Methods for Convection � Di usion Problems on General Grids
2002
This paper summarizes a number of discretization and solution tech niques that are commonly referred to as nite volume and mixed nite element methods We describe some of them in more detail that is provide a common discretization strategy as well as address the issue of solving the resulting systems of linear algebraic equations The paper focuses on locally conservative discretizations that include both unstruc tured and non matching grids
Approximation of parabolic problems on grids locally refined in time and space
Applied Numerical Mathematics, 1994
We present a strategy for solving time-dependent problems on grids with local re nement in time and in space, using di erent time-space sizes in di erent regions of space. We discuss two approximations based on the discontinuous Galerkin method and the nite di erence method with piecewise constant and piecewise linear interpolation in the time direction along the interface between the coarse-and negrid regions. Next, we present an iterative method for solving the composite-grid system that is based on domain decomposition techniques and reduces to solution of standard problems with standard time stepping (alternatively on the coarse and ne grids). Finally, numerical results that con rm both the analysis and the convergence theory of the iterative method are presented.
Analysis of finite element schemes for convection-type problems
International Journal for Numerical Methods in Fluids, 1995
Various finite element schemes of the Bubnov–Galerkin and Taylor–Galerkin types are analysed to obtain the expressions of truncation errors. This way, dispersion errors in the transient, and diffusion errors both in the transient and in the steady state, are identified. Then, with reference to the transient advection–diffusion equation, stability limits are determined by means of a general von Neumann procedure. Finally, the operational equivalence between Taylor–Galerkin methods, utilized for pseudo-transient calculations, and Petrov–Galerkin methods, derived for the steady state forms of the advection–diffusion equation, is illustrated. Theoretical conclusions are supported by the results of numerical experiments.
A Comparison of Time Discretization Schemes in the Solution of a Convection-Diffusion Equation
2011
The aim of this work is to present in a didactic way a comparison of analytical and numerical solutions of a transient one-dimensional heat transfer problem. So, the solution of a parabolic equation of convection-diffusion type has been done with discretization in space by the Central Finite Difference method and in the time by five schemes: two explicit and three implicit. From the analytical and the numerical results the L2 norm and the L-infinity norm are evaluated as measures of the error. Some numerical applications are presented and discussed.
Applications of Composite Grid Method for Free-Surface Flow Computations by Finite Difference Method
Journal of the Society of Naval Architects of Japan, 1994
The paper presents some new numerical investigations of the free-surface viscous flow around a submerged NACA 0012 hydrofoil based on the 2-D laminar incompressible Navier-Stokes equations by using the finite difference method where a composite grid system is applied. A composite grid technique is proposed for the free-surface flow which has two parameters that describe the phenomenon : Reynolds number around the body and the Froude number. Because the latter is equivalent to the wave number or the wave length, the grid generation must meet the two different requirements according to these parameters. The physical domain is divided into several subdomains that are covered by different grids according to the main flow characteristics. Since the boundary condition is a Dirichlet-type one for all the component grids, the grids must overlap and the communication between them is realised by interpolation. The proposed numerical method is based on a Schwartz iterative procedure. The paper presents considerations concerning both interpolation and iteration processes. The results obtained by the present numerical scheme are compared with those obtained by the monoblock grid and show a good agreement. The finite difference scheme is based on the use of the Euler-type kinematic boundary condition at the free-surface. The third order upwind difference scheme with the third derivative of the wave elevation is employed. The formulation is of a higher order of accuracy than those usually used where the position of the particle is determined locally. It takes into account the influence of the neighbouring particle movements, thus preserving the characteristics of the wavy motion. As a result, a faster elevation is obtained. The flow still oscillates at the downstream and the zero-extrapolation is only valid for the viscous diffusion. Therefore an added dissipation zone is introduced as boundary condition at the downstream. Inside it waves are numerically damped within a certain range. As a result, instabilities determined by the numerical wave reflection are avoided.
Six numerical schemes for parabolic initial boundary value problems with a priori bounded solution
The paper presents six new numerical finite-difference schemes to solve nonlinear parabolic initial boundary value problems with constraints imposed a priori on the solution. The first three proposed schemes are based on a special first-order approximation of the "diffusion" and "transport" terms combined with an unconditionally stable Gauss-Seidel-type iterative technique. We present a theoretical analysis of the method as applied to the diffusion wave equation and to the generalized diffusion wave approach. The fourth scheme allows for an accuracy enhancement at the expense of the computational cost. The scheme employs an adaptive grid which guarantees the second-order approximation or nearly so. An efficiency of the proposed techniques is demonstrated by the numerical simulations of flood in the eastern areas of Bangkok. Next, the first-order techniques are generalized to the Richard's type model of unsaturated porous medium flows, representing the general case of a nonlinear parabolic equation endowed with the space dependent bilateral constraints. The fifth and the sixth schemes are designed for space independent and space dependent constraints, respectively. We present a theoretical analysis of the methods. Finally, we verify the proposed schemes by methodological applications and analyze the convergence rate.
International Journal for Numerical Methods in Engineering, 1972
The convergence and accuracy characteristics of two commonly used finite difference schemes, namely the central and the upwind differences, are compared with a scheme recently proposed by Spalding. For a simple two-dimensional problem with conduction and convection, it is shown that the scheme proposed by Spalding is a preferable choice both from the point of view of accuracy and that of convergence. The extension of this conclusion to a general case must await further substantiation.
A collocation method for the solution of convection – diffusion parabolic problems
2012
on the domain D , where D T , 0 1 , 0 , with initial condition u( 0 , x ) = ) ( 0 x u , x(0,1) , (2) and boundary donditions u(0,t) = ) ( 0 t g , u(1,t) = ) ( 1 t g , t(0,T] , (3) We assume that 1. functions c(x,t) , p(x,t) and f(x,t) are sufficiently smooth on the D ,and c(x,t) 0 , p(x,t) 0 0 p ,(x,t) D 2. functions ) ( 0 t g and ) ( 1 t g are sufficiently smooth on the T , 0 and ) (x uo is smooth on 1 , 0 , 3. 1 , 0 ,and 4. Compatibility conditions are satisfied at the corner points 0 , 0 and 0 , 1 . We suppose that 2 , ) , ( ) , ( ) , , , ( t x f u t x c u u t x x ) , , , ( ) , ( x t xx u u t x u t x p u In this paper we have developed two-level implicit difference scheme by using cubic B-spline function for the solution of singularly perturbed parabolic problem (1). This paper is arranged as follows. In section2, we present a finite difference approximation to discretize the equation (1) and obtain the conver...