Showing papers on "Finite difference published in 1982"

Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures*

Abstract: Finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form $cu_t + bu_x - (au_x )_x = f$. Optimal order error estimates in $L^2 $ and $W^{1,2} $ are derived for the finite element procedure. Various error estimates are presented for a variety of finite difference methods. The estimates show that, for convection-dominated problems $(b \gg a)$, these schemes have much smaller time-truncation errors than those of standard methods. Extensions to n-space variables and time-dependent or nonlinear coefficients are indicated, along with applications of the concepts to certain problems described by systems of differential equations.

Upwind difference schemes for hyperbolic systems of conservation laws

Abstract: We derive a new upwind finite difference approximation to systems of nonlinear hyperbolic conservation laws. The scheme has desirable properties for shock calculations. Under fairly general hypotheses we prove that limit solutions satisfy the entropy condition and that discrete steady shocks exist which are unique and sharp. Numerical examples involving the Euler and Lagrange equations of compressible gas dynamics in one and two space dimensions are given.

Introduction to Groundwater Modeling: Finite Difference and Finite Element Methods

Abstract: Introduction to Groundwater Modeling presents an overview of the fundamental concepts and applications of computerized groundwater modeling.

Why Particle Methods Work

Abstract: The theme of this paper is that particle methods are closely related to both finite difference and spectral methods because the three methods can be considered special cases of interpolation by kernel estimation. The kernels for a number of special cases are given in detail, and the accuracy of the resulting interpolation is analyzed. A general procedure for deriving equations for numerical work from the equations of hydrodynamics is described. It is applied to the derivation of the SPH equations which conserve linear and angular momentum exactly.

Determination of source parameters by wavefield extrapolation

Abstract: Summary The extrapolation of an observed (seismogram) wavefield backward in time produces an image, in both time and space, of the source. This concept is implemented through a finite difference solution of the two-dimensional acoustic wave equation. The seismograms themselves are used as time-dependent boundary values that drive the finite difference mesh, Numerical examples illustrate the determination of hypocentre location and origin time for point sources and of source extent, orientation and rupture velocity for finite sources in both homogeneous and heterogeneous media. Spatial resolution of source parameters is typically one-half wavelength of the dominant frequency generated by the source.

A More Accurate Finite Difference Approximation for the Valuation of Options

Abstract: option when a closed-form solution of the valuation equation cannot be ob? tained. This model is based on a difference approximation of the valuation equation and uses standard numerical methods. We intend to show here that the same methods can be used to derive a difference approximation of the solution of the valuation equation which has a greater level of accuracy than Schwartz's approximation. In the first section, we will show why the difference approximation that we will use is more accurate. We will also illustrate this greater level of accuracy by applying both the new difference approximation and Schwartz's algorithm to the valuation of a call option on a nondividend paying stock and by comparing the results obtained to the closed-form solution of Black and Scholes [1]. Finally, in Section ll,we will show that the algorithm proposed in Sec? tion I suppresses the bias that Brennan and Schwartz [2] found in the variance of the generalized jump process which approximates the diffusion process followed by the logarithm of the stock price. Conclusions also will be presented in this section.

Efficient Sequential Solution of the Nonlinear Inverse Heat Conduction Problem

Abstract: The nonlinear inverse heat conduction problem is the calculation of surface heat fluxes and temperatures by utilizing measured interior temperatures in opaque solids possessing temperature-variable thermal properties. The most widely used numerical method for this problem was developed by Beck. The new sequential procedure presented here reduces the number of computer calculations by a factor of 3 or 4. The general heat conduction model used permits treatment of various one-dimensional geometries (plates, cylinders, and spheres), energy sources, and fin effects. The numerical procedure is illustrated for finite differences, but the basic concepts are also applicable to the finite-element method. Detailed descriptions of the computational algorithms are given and a nonlinear example is provided.

Finite difference methods and their convergence for a class of singular two point boundary value problems

Abstract: We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (x?y?)?=f(x,y), y(0)=A, y(1)=B, 0

The ρ‐family of algorithms for time‐step integration with improved numerical dissipation

Abstract: The dynamic analysis of complex non-linear structural systems by the finite element approach requires the use of time-step algorithms for solving the equations of motion in the time domain. Both an implicit and an explicit version of such a time-step algorithm, called the ρ-method, the parameter ρ being used for controlling numerical damping in the higher modes, are presented in this paper. For the implicit family of algorithms unconditional stability, consistency, convergence, accuracy and overshoot properties are first discussed and proved. On the basis of the algorithmic damping ratio (dissipation) and period elongation (dispersion) the ρ-method is then compared with the well-known implicit algorithms of Hilber, Newmark, Wilson, Park and Houbolt. An explicit version of the algorithm is also derived and briefly discussed. This shows numerical properties similar to the central difference method. Both versions of the algorithm have been implemented in a general purpose computer program which has been often used for both numerical tests and practical applications.

Finite-Difference Approximations and Superconvergence for the Discrete-Ordinate Equations in Slab Geometry

Abstract: A unified framework is developed for calculating the order of the error for a class of finite-difference approximations to the monoenergetic linear transport equation in slab geometry. In particular, the global discretization errors for the step characteristic, diamond, and linear discontinuous methods are shown to be of order two, while those for the linear moments and linear characteristic methods are of order three, and that for the quadratic method is of order four. A superconvergence result is obtained for the three linear methods, in the sense that the cell-averaged flux approximations are shown to converge at one order higher than the global errors.

On the Numerical Solution of Initial/Boundary-Value Problems in One Space Dimension

Abstract: The numerical solution of initial/boundary-value problems of the form \[ A(u,x,t)u_t + B(u,x,t)u_x = c(u,x,t)\] is considered. Particular emphasis is placed on the solution of problems with large gradients, e.g., shocks and boundary layers. A mesh-selection technique is described that accurately places points in regions where the solution is rapidly changing. This is accomplished by a transformation of the original equations to a new coordinate system. Finite difference solutions for two sample problems are calculated.

Aspects of Numerical Methods for Elliptic Singular Perturbation Problems

Abstract: Upwind difference, defect correction and central difference schemes for the solution of the convection-diffusion equation with small viscosity coefficient are compared. It is shown that central difference schemes and hence also standard Galerkin finite element methods are preferable above upwind and defect correction schemes, when Gaussian elimination is used for the solution of the resulting system of equations.When iterative solution methods are employed good results can be achieved by a defect-correction method, whereas upwind difference schemes are generally inaccurate.

A finite‐difference treatment of interface conditions for the parabolic wave equation: The irregular interface

Abstract: A finite‐difference solution to the parabolic wave equation is extended to treat an irregular interface. Interface conditions are developed to preserve continuity of pressure and continuity of the normal component of particle velocity at the boundary between media having different sound speeds and densities. A complete mathematical treatment for the case of an irregular interface is presented. To demonstrate the method, numerical results are obtained for sound propagation from deep to shallow water.

Steady flow in a sudden expansion at high Reynolds numbers

Abstract: The sudden expansion of a laminar flow in a two‐dimensional channel is examined theoretically in the limit of large Reynolds number R. Previous investigators found, from experiment and from numerical solutions of the equations of motion, that a region of closed streamlines is formed whose streamwise length is linearly related to R for R = O(102). It is desired to determine if the steady solutions to the Navier–Stokes equations continue to exhibit this relationship indefinitely for increasing R. Since solutions are sought for which the longitudinal length scale is O(R) and that in the tranvserse direction is O(1), the equations of motion reduce to the boundary‐layer equations as R→∞. These equations are solved numerically using a finite difference technique for selected values of λ, the ratio of the upstream channel half‐width to the step height. Steady solutions are found for all values of λ when the inlet velocity profile is parabolic. However, a uniform inlet velocity profile yields steady solutions wit...

Collocation for Singular Perturbation Problems III: Nonlinear Problems without Turning Points

Abstract: A class of nonlinear singularly perturbed boundary value problems is considered, with restrictions which allow only well-posed problems with possible boundary layers, but no turning points. For the numerical solution of these problems, a close look is taken at a class of general purpose, symmetric finite difference schemes arising from collocation. .br It is shown that if locally refined meshes, whose practical construction is discussed, are employed, then high order uniform convergence of the numerical solution is obtained. Nontrivial examples are used to demonstrate that highly accurate solutions to problems with extremely thin boundary layers can be obtained in this way at a very reasonable cost.

Prediction of rough-wall skin friction and heat transfer

Abstract: A finite difference solution to the boundary-layer equations for flow over rough surfaces is presented. The boundary-layer equations are cast in a form to account for the blockage effects of roughness elements. The roughness effect is described by a sink term in the momentum equation and a source term in the static enthalpy equation. A two-layer algebraic mixing length model that accounts for low Reynolds numbers, surface roughness, and wall transpiration is developed. Good agreement is shown for several comparisons to rough- wall, flat-plate and sharp-cone data.

Finite Elements With Characteristics for Two-Component Incompressible Miscible Displacement

Abstract: An efficient method for modeling convection-dominated flows is presented and applied to miscible displacement in a porous medium. The method uses characteristics to model convection and finite elements for diffusion and dispersion, thereby treating each physical process with a well-suited numerical scheme. A finite difference analogue can also be formulated. Numerical results are given that show that the method can simulate adverse mobility ratio displacements accurately without suffering from grid orientation, numerical dispersion, or overshoot. 14 refs.

IFD: An Implicit Finite-Difference Computer Model for Solving the Parabolic Equation

Abstract: : A general purpose computer model, based on the implementation of an implicit finite-difference scheme, is developed for the solution of the parabolic wave equation. This model can be used to predict acoustic propagation loss in both range-independent and range-dependent environments. An important feature of the model is that it can handle arbitrary surface boundary conditions and an irregular bottom with arbitrary bottom boundary conditions. In the event that the bottom boundary conditions cannot be expressed mathematically, the model has the capability of introducing an artifical absorbing bottom such that, with the appropriate bottom attenuation, the problem becomes solvable. Another important feature of the model is that it can handle horizontal interfaces of layered media. The model is easy to use and easy to modify. Numerical test examples are included to demonstrate the capabilities of the model. (Author)

A vortex wake capturing method for potential flow calculations

Abstract: A method is presented for modifying finite difference solutions of the potential equation to include the calculation of non-planar vortex wake features. The approach is an adaptation of Baker's 'cloud in cell' algorithm developed for the stream function-vorticity equations. The vortex wake is tracked in a Lagrangian frame of reference as a group of discrete vortex filaments. These are distributed to the Eulerian mesh system on which the velocity is calculated by a finite difference solution of the potential equation. An artificial viscosity introduced by the finite difference equations removes the singular nature of the vortex filaments. Computed examples are given for the two-dimensional time dependent roll-up of vortex wakes generated by wings with different spanwise loading distributions.