Numerical Methods for Time-Dependent, Nonlinear Boundary Value Problems

Culham, W.E.; Varga, Richard S.

doi:10.2118/2806-pa

Cited by 38 publications

(12 citation statements)

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“…Linearized (noniterative) variants of Picard and Newton methods can be derived from a single iteration of these schemes with ψ n as the initial estimate [ Paniconi et al , 1991]. For example, the noniterative “modified implicit Euler” scheme [ Culham and Varga , 1971] can be formulated as When expressed in this form, the modified implicit Euler scheme can be supplemented with the adaptive truncation error control described above. However, Paniconi et al [1991] showed that when applied to O (Δ t 2 ) formulations, e.g., the Crank‐Nicolson scheme, linearizations based on ψ n reduce the accuracy to O (Δ t ).…”

Section: Numerical Formulationmentioning

confidence: 99%

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Noniterative time stepping schemes with adaptive truncation error control for the solution of Richards equation

Kavetski

Binning

Sloan

2002

Water Resources Research

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[1] Noniterative implicit time stepping schemes with adaptive temporal truncation error control are developed for the solution of the pressure form of Richards equation. First-and second-order linearizations of an adaptive backward Euler/Thomas-Gladwell formulation are introduced and are shown to constrain the temporal truncation errors near a userprescribed tolerance and maintain adequate mass balance. Numerical experiments demonstrate that accurate noniterative linearizations achieve cost-effective solutions of problems where soils are described by highly nonlinear and nonsmooth constitutive functions. For these problems many conventional iterative solvers fail to converge. The noniterative formulations are considerably more efficient than analogous time stepping schemes with iterative solvers. The second-order noniterative scheme is found to be more efficient than the first-order noniterative scheme. The proposed adaptive noniterative algorithms can be easily incorporated into existing backward Euler software, which is widely used for Richards equation and other nonlinear PDEs.

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Section: Numerical Formulationmentioning

confidence: 99%

“…[18] Linearized (noniterative) variants of Picard and Newton methods can be derived from a single iteration of these schemes with y n as the initial estimate [Paniconi et al, 1991]. For example, the noniterative ''modified implicit Euler'' scheme [Culham and Varga, 1971] can be formulated as…”

Section: Noniterative Linearization Of the Implicit Systemmentioning

confidence: 99%

Noniterative time stepping schemes with adaptive truncation error control for the solution of Richards equation

Kavetski

Binning

Sloan

2002

Water Resources Research

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“…In this method first the grid block sizes~Xi = 1, 2, ... , n., are arbitrarily selected; and then, keeping the first grid point at the center of the first grid block, the second grid point is located at the position where 6x; = 6xt, the third grid point is located at the position where 6x; = 6xt, and in general (7) (8) …”

Section: Pseudopoint-distributed (Methods 4)mentioning

confidence: 99%

Use of Irregular Grid in Reservoir Simulation

Nacul¹,

Aziz²

1991

Proceedings of SPE Annual Technical Conference and Exhibition

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In order to numerically solve the system of nonlinear partial differential equations which describes multiphase flow in porous media, finite volume discretization in space and finite-difference in time are commonly used in reservoir simulation 1 • This requires the calculation of flow at the boundaries between adjacent blocks using the pressure information at a specific point within each block. Consequently, the knowledge of both, block boundaries and grid point locations are required. Therefore, the distribution of grid points in space and the location of block boundaries with respect to grid points influence the accuracy of finite-difference approximations.In this paper, we investigate the accuracy of six different kinds of gridding systems, which are derived from the classical point-dis~ributed and the block-centered methods. Our goal is to obtain maximum accuracy for a given number of blocks. To achieve this goal, two approaches are studied:1. Given a set of grid points, decide where to place the block boundaries.

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“…7.29(10r2 (1.00) 2~6 1.49(10)-'3 (-0.69) 1.49(10)-7 (3.00) 1.14(10)"* (2.00) 3.64(10)"2 (1.00) Table 3 Error Norms for Table 4 Error Norms for Problem 4 e(t;h)\\' \\e'(t; h)\\' \\e"(t; h)\\' \\e"'(t; h)\\' 1 2.27(10)° 1.15(10)' 1.48(10)3 5.59(10)3 2"' 3.45(10)"' (6.04) 1.80 (10) OH* !. ; (01 -U-.…”

Section: The Galerkin Methods As An Implicit Runge-kutta Method Wrighmentioning

confidence: 99%

“…Finally, we remark that recent "semidiscrete" Galerkin methods [7], [9Jj [18], [19], [23] reduce initial-boundary value problems to systems of ordinary differential equations. When combined with such methods, our techniques open the possibility of "fully discrete" Galerkin methods for these problems.…”

Section: Introductionmentioning

confidence: 99%

One-Step Piecewise Polynomial Galerkin Methods for Initial Value Problems

Hulme¹

1972

Mathematics of Computation

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Abstract. A new approach to the numerical solution of systems of first-order ordinary differential equations is given by finding local Galerkin approximations on each subinterval of a given mesh of size h. One step at a time, a piecewise polynomial, of degree n and class C°, is constructed, which yields an approximation of order 0(A*") at the mesh points and 0(A"+1) between mesh points. In addition, the y'th derivatives of the approximation on each subinterval have errors of order 0(An_'+1), 1 £ j £ n. The methods are related to collocation schemes and to implicit Runge-Kutta schemes based on Gauss-Legendre quadrature, from which it follows that the Galerkin methods are /4-stable.1. Introduction.In this paper, we show how Galerkin's method can be employed to devise one-step methods for systems of nonlinear first-order ordinary differential equations. The basic idea is to find local nth degree polynomial Galerkin approximations on each subinterval of a given mesh and to match them together continuously, but not smoothly.For each /i _ 1, a method is defined (Section 2) which uses an n-point GaussLegendre quadrature formula to evaluate certain inner products in the Galerkin equations. For sufficiently small step size h, a unique numerical solution exists and may be found by successive substitution (Section 3). After showing that these Galerkin methods are also collocation methods (Section 4) and implicit Runge-Kutta methods (Section 5), we show that the mesh point errors are of the order 0(h2n), and the global errors are of the order 0(hn+1) for the approximate solution and 0(h"~'+1), 1 ^ j ^ n, for its yth derivatives (Section 6). A proof of the ^-stability of the methods is given in Section 7, and numerical results are presented in Section 8.Discrete one-step methods based on quadrature, other than the classical RungeKutta methods, have been studied by several authors, including the explicit schemes

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Numerical Methods for Time-Dependent, Nonlinear Boundary Value Problems

Abstract: This paper presents and examines in detail and computer core storage required. These tests indicate that the higher-order Galerkin methods require the [east amount of computer time for a given range of accuracy.

Cited by 38 publications

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Noniterative time stepping schemes with adaptive truncation error control for the solution of Richards equation

Noniterative time stepping schemes with adaptive truncation error control for the solution of Richards equation

Use of Irregular Grid in Reservoir Simulation

One-Step Piecewise Polynomial Galerkin Methods for Initial Value Problems

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