Boundary value method

Solution of the equations in (2.8-2.18) proceeds with an adaptive nonlinear boundary value method. The solution procedure has been discussed in detail elsewhere (10) and we outline only the essential features here. Our goal is to obtain a discrete solution of the governing equations on the mesh Af... 

Boundary value methods provide a description of the solution either by providing values at specific locations or by an expansion in a series of functions. Thus, the key issues are the method of representing the solution, the number of points or terms in the series, and how the approximation converges to the exact answer, i.e., how the error changes with the number of points or number of terms in the series. These issues are discussed for each of the methods finite difference, orthogonal collocation, and Galerkin finite element methods. 

M.D. Smooke. Solution of Burner-Stabilized Premixed Laminar Flames by Boundary-Value Methods. J. Comp. Phys., 48 72-105,1982. 

M.D. Smooke, J.A. Miller, and RJ. Kee. Determination of Adiabatic Flame Speeds by Boundary Value Methods. Comb. Sci. Techn., 34 79-89,1983. 

Figure 3.23 Drawing composition profiles by the boundary value method.

A computational approach of a very different nature, usually referred to as the boundary value method , introduces a multidimensional finite-difference mesh to directly solve the partial differential equations of reactive scattering This approach has been taken by Diestler and McKoy (1968) in work related to a previous one by Mortensen and Pitzer (1962). While the second authors used an iterative procedure to impose the physical boundary conditions of scattering, the more recent work constructs the wavefunction as a linear combination of independent functions Xj which satisfy the scattering equation for arbitrarily chosen boundary conditions. 

The above-mentioned computational technique has been compared with the finite difference boundary-value method as modified by Truhlar and Kupperman (Diestler et ah, 1972) for a Wall-Porter fit to the Shavitt et al. (1968) surface. Results over the total energy range 0-014 a.u. to 0-400 a.u. agree typically to 1 %, except for some very small probabilities. 

Smooke, M. D., "Solution of Burner Stabilized Pre-Mixed Laminar Flames by Boundary Value Methods," Sandia National Laboratories Report 81-8040 (1982). 

The "finite difference boundary value" method was recently used to compute the transition probabilities for the colinear reaction... 

L. Brugnano and D. Trigiante, Convergence and stability of boundary value methods for ordinary differential equations, J. Comput. Appl. Math., 1996, 66(1-2), 97-109. 

L. Brugnano and D. Trigiante, Solving Differential Problems by Multistep Initial and Boundary Value Methods, Gordon and Breach Science Publishers, Amsterdam, 1998. 

We solve equations (17) and (20) using the finite difference boundary value method... 

The reflection and transmission properties of multiple layers of materials with different refractive indices can be treated either as a ray tracing or as a boundary value problem (e.g., Wolter, 1956 Bom Wolf, 1975). The ray tracing method leads to summations where it is sometimes difficult to follow the phase relations, especially if several layers are to be treated. We follow closely the boundary value method reviewed by Wolter (1956). In effect this method is a generalization of the one-interface boundary problem that led to the formulation of the Fresnel equations in Section 1.6. 

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