Peaceman-Rachford

The wave function is propagated forward in time via a Peaceman-Rachford alternating-direction, implicit scheme [6] given by... 

Here, the matrices H and V are symmetric and positive definite matrices, which are each, after suitable permutation of indices, tridiagonal matrices. The matrix S is a non-negative diagonal matrix. Recalling that tridiagonal matrix equations are efficiently solved by the Gauss elimination method, we consider now the Peaceman-Rachford iterative method [27], a particular variant of the lAD methods, which is defined by... 

Theorem 4. If (4.26) is valid, then the Euclidean norm of the error vector cifter m applications of the Peaceman-Rachford process (4.24)-(4.24 ) satisfies... 

Theorem 5. If H and V are symmetric and positive definite matrices, and 2 is a non-negative definite matrix, then the Peaceman-Rachford matrix Tp is convergent for any fixed p > 0. 

For purposes for comparison, we again consider the numerical solution of the Dirichlet problem for the unit square with uniform mesh spacings A = jp. For this problem, it can be shown that (4.26) is valid, and that the average rate of convergence R[T p. ] of the Peaceman-Rachford iterative method, with suitably selected parameters pu is [9 56]... 

The Peaceman-Rachford ADI method is second-order with respect to time, and performs similarly to Crank-Nicolson. Indeed, Lapidus and Finder write [224, p. 246] ... is a variation of the Crank-Nicolson approximation . It is known to be unconditionally stable [225]. As with CN, ADI may show some error oscillations, as also evidenced by the fact that some habitually use expanding time intervals when employing ADI [226-231], although some of these same workers on occasion also use equal time intervals [232,233]. 

Hunsdorfer WH, Verwer JG (1989) Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems. Math Comput 53 81-101... 

The scheme ascribed to Peaceman and Rachford provides some realization of this idea and refers to implicit alternating direction schemes. The present values y = and y = of this difference scheme are put... 

The scheme ascribed to Peaceman and Rachford provides some realization of this idea and refers to implicit alternating direction schemes. The present values y = yn and y = yn+1 of this difference scheme are put together with the intermediate value y = j/n+1 2, a formal treatment of which is the value of y at moment t = tn+]/2 = tn + r/ 2. The passage from the nth layer to the (n + 1)th layer can be done in two steps with the appropriate spacings 0.5 r ... 

Bruce GH, Peaceman DW, Rachford HH, Rice JD (1953) Trans Am Inst Min Engrs 198 79... 

Finally, there is another mode of operation for ees. If one considers the total simulation time as a single step, this can be subdivided into a number of exponentially expanding subintervals in the same manner as the above description of subdivision of the first interval. This was first suggested by Peaceman and Rachford in 1955 [436], in their famous paper describing the ADI method (see Chap. 12), and was used later [236,355]. It is routinely used by Svir and coworkers [538, 540]. These workers tend to use strong expansion with 7 = 2, which has been found not to be optimal [149], The method requires a large number of recalculations of the coefficients and thus uses more computer time than equal intervals with a damping device applied to the first interval. 

The ADI method, first used by Peaceman and Rachford (1955) for solving parabolic PDEs, can also be derived from the Crank-Nicholson algorithm. In the three-dimensional MRTM model, the governing equation can be discretized by the Crank-Nicholson algorithm as... 

Peaceman, D.W. and Rachford, H.H., The numerical solution of parabolic and elliptic differential equations, SIAM J., 3, 28, 1955. 

The alternating direction implicit (ADI) method (Peaceman and Rachford, 1955) is a partially implicit method. The equation is rearranged so that one coordinate may be solved implicitly using the Thomas algorithm whilst the others are treated explicitly. If this is done alternately, each coordinate has a share of the implicit iterations and the efficiency (Gavaghan and Rollett, 1990) as well as the stability is improved. The method was used by Heinze for microdisc simulations (Heinze, 1981 Heinze and Storzbach, 1986) and has subsequently been adopted by others (Taylor et al, 1990 Fisher et al., 1997). 

This equation is to be formulated for all grid points (i, j). A system of linear equations for the unknown temperatures at time tk+l, that has to be solved for every time step, is obtained. Each equation contains five unknowns, only the temperature at the previous time tk is known. A good solution method has been presented by P.W. Peaceman and H.H. Rachford [2.69]. It is known as the alternating-direction implicit procedure (ADIP). Here, instead of the equation system (2.305) two tridiagonal systems are solved, through which the computation time is reduced, see also [2.53]. 

The heat transfer in Eq. 6.3 is solved by implicit finite differences. This solution closely resembles that of Peaceman and Rachford (i955)> but is modified to include the variable values of the thermophysical parameters and variable steps dZ ... 

Peaceman DW, Rachford HH (1955) The numerical solution of parabolic and elliptic differential equations. Journal of Society of Industrial and Applied Mathematics 3(i) 28-4i Perrier B, Quiblier J (1974) Thickness changes in sedimentary layers during compaction history methods for quantitative evaluation. AAPG Bulletin 58(3) 507-520 Perry EA Jr, Hower J (1970) Burial diagenesis in Gulf Coast pelitic sediments. Clays and Clay Minerals 16 15-30... 

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