Von Neumann boundary condition

Application of ABC of Eqs. (12.19), (12.23), and (12.25), on the one side, and standard Dirichlet or von Neumann boundary conditions at open boundaries, on the other side, reveals the drastic effect of outlet boundary conditions on the flow pattern. 

The necessity to solve Laplace s equation requires formulating all boundary conditions, and at this point the cell geometry becomes important. Generally, there are two types of boundary conditions that come into play. Any electrically insulating cell wall is mathematically described by zero-flux or von Neumann boundary conditions ... 

Boundary value — A boundary value is the value of a parameter in a differential equation at a particular location and/or time. In electrochemistry a boundary value could refer to a concentration or concentration gradient at x = 0 and/or x = oo or to the concentration or to the time derivative of the concentration at l = oo (for example, the steady-state boundary condition requires that (dc/dt)t=oo = 0). Some examples (dc/ dx)x=o = 0 for any species that is not consumed or produced at the electrode surface (dc/dx)x=o = -fx=0/D where fx=o is the flux of the species, perhaps defined by application of a constant current (-> von Neumann boundary condition) and D is its diffusion coefficient cx=o is defined by the electrode potential (-> Dirichlet boundary condition) cx=oo, the concentration at x = oo (commonly referred to as the bulk concentration) is a constant. 

Neumann (von ) boundary condition - von Neumann boundary condition... 

Figure 12.2 Application of the nonreflecting boundary conditions (left part) and standard von Neumann boundary conditions (right part) in the problem on pressure disturbance propagation in a flow reactor with open left and right boundaries. Time instants (a) 10 /is, (b) 20 /is, (c) 30 /is. Flow velocity at the inlet 40 m/s, po = 0.1 MPa, To = 300 K, ko = 9 J/kg, lo = 2 mm. Initial pressure differential Ap/po = 0.5. The size of the computational domain is 3.3 x 2 cm...

F.M. Fernandez, E.A. Castro, Hypervirial analysis of enclosed quantum mechanical systems. II. von Neumann boundary conditions and periodic potentials, Int. J. Quant. Chem. 19 (4) (1981) 533-543. 

When solving diffusion equations it is common to use second-order accurate approximations, so that simply setting cpo = (p is not file preferred way to treat a von Neumann boundary condition. Rather, we obtain second-order accuracy by fitting a quadratic polynomial to (piX) near f = 0,... 

The discretized form of the von Neumann boundary condition is then... 

Previously, when solving the Poisson equation with Dirichlet boundary conditions, we obtained a matrix that was positive-definite and could be solved witti the conjugate gradient method. For this problem, however, we have a number of von Neumann boundary conditions, e.g. at the grid points, (x , = 0, ft), for which an approximation of the boundary... 

The von Neumann method described above usually works well, and is reasonably easy to apply. One reason it works well, despite the fact that it totally ignores conditions at the boundaries, is that errors that often arise at interior points away from the boundaries and spread from there [private communication with O. Osterby 1996], However, boundary conditions can affect stability, especially if derivative (or mixed) boundary conditions hold [116,117,118,119,334], It might be safer to consider all points in space in some way. The following somewhat brief treatment is described in greater mathematical detail in such texts as Smith [514] or Lapidus and Pinder [350],... 

Of the statistical simulations, two major types are distinguished cellular automata (CA) and Monte Carlo (MC) simulations. The basic ideas concerning CA go back to Wiener and Rosenblueth [1] and Von Neumann [2]. CA exist in many variants, which meikes the distinction between MC and CA not always clear. In general, in both techniques, the catalyst surface is represented by a matrix of m x n elements (cells, sites) with appropriate boundary conditions. Each element can represent an active site or a collection of active sites. The cells evolve in time according to a set of rules. The rules for the evolution of cells include only information about the state of the cells and their local neighborhoods. Time often proceeds in discrete time steps. After each time step, the values of the cells are updated according to the specified rules. In cellular automata, all cells are updated in each time step. In MC simulations, both cells and rules are chosen randomly, and sometimes the time step is randomly chosen as well. Of course, all choices have to be made with the correct probabilities. 

The von Neumann stability analysis apphes ordy to difference equations with constant coefficients and periodic boundary conditions. Instability may arise from specification of boundary conditions or nonlinear terms in differential equations. Instability caused by nonlinear terms is called nonlinear instability and was first noticed in a numerical solution of the nonhnear barotropic vorticity equationinthe early days of munerical weatherprediction. 

Modern computer-based numerical analysis really got started with the 1947 paper by John von Neumann and Herman Goldstine, Numerical Inverting of Matrices of High Order, which appeared in the Bulletin of the American Mathematical Society. Following that, many new and improved techniques were developed for numerical analysis including cubic-spline approximation, sparse-matrix packages, and the finite element method for elliptic PDEs with boundary condition. 

Numerical stability. The stability of a scheme can be determined by a relatively simple von Neumann stability test (Carnahan, Luther, and Wilkes, 1969 Richtmyer and Morton, 1957). This test, we emphasize, is only qualitatively accurate, since it does not account for the detailed effects of boundary and initial conditions, and for the role of heterogeneities when variable coefficients are present in the given equation. Note that a stable finite difference scheme does not necessarily converge to solutions of the PDE even if Ar, At are vanishingly small. This subject is treated in advanced courses. 

We next extend the finite difference method to treat BVPs of greater complexify, with non-Cartesian coordinates and nommiform grids, von Neumann-type boundary conditions, multiple fields, time dependence, and PDFs in more than two spatial dimensions. We do so through the examples in the following sections. 

This BVP introduces several new issues (1) nonCartesian (spherical) coordinates, (2) more than one coupled PDE, and (3) a BC at r = 0 that specifies the local value of the gradient (a von Neumann-type boundary condition). Also, experience tells us that when internal mass transfer resistance is strong, reaction only occurs within a thin layer near the surface over which the local concenUation of A drops rapidly to zero. Thus, we use a computational... 

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