Numerical Methods of Higher-Order Accuracy for Diffusion- Convection Equations

Price, H.S.; Cavendish, James C.; Varga, Richard S.

doi:10.2118/1877-pa

Cited by 109 publications

(41 citation statements)

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“…*) = 0, 1 =■ A: g n, and let / = q' so that w' = /, «(*,-) = 1 leads to u = q = y on [r,, r<+l]. Substituting y = fl and / = fl' into (17) yields (18) at = 0, 1 Jg * £ b.…”

Section: Piecewise Polynomial Galerkin Methodsmentioning

confidence: 99%

One-step piecewise polynomial Galerkin methods for initial value problems

Hulme¹

1972

Math. Comp.

124

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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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“…*) = 0, 1 =■ A: g n, and let / = q' so that w' = /, «(*,-) = 1 leads to u = q = y on [r,, r<+l]. Substituting y = fl and / = fl' into (17) yields (18) at = 0, 1 Jg * £ b.…”

Section: Piecewise Polynomial Galerkin Methodsmentioning

confidence: 99%

One-step piecewise polynomial Galerkin methods for initial value problems

Hulme¹

1972

Math. Comp.

124

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“…Alternatively, a general purpose stiff ordinary differential equation solver could be used on (3), but Byrne and Hindmarsh' suggest that specially analyzed methods such as those described below are more suitable. Finite element approximation can also lead to continuous time discrete space representations of (l), for example through Galerkin's method, see Price et al, 3 and Smith et aL4 The equations thus derived resemble (3) but have the form with an extra matrix appearing on the left hand side. Crank-Nicolson treatment in time yields TIME DISCRETIZATIONS Alternative discretizations to (4) or (6) in the solution of (3) or ( 5 ) will now be considered.…”

Section: Au Ax U ( X O ) = O U ( O T ) = 1-(lt)=omentioning

confidence: 99%

Evaluation of Nørsett methods for integrating differential equations in time

Smith

Siemieniuch

Gladwell

1977

Num Anal Meth Geomechanics

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SUMMARYSome recently developed implicit time discretizations are discussed whose main application is to solving ordinary differential equations arising from finite element approximations to partial differential equations. Their theoretical properties, computer implementation and numerical behaviour, as observed in tests on simple examples, are compared with well-known discretizations such as the Crank-Nicholson method. INTRODUCI'IONIn the numerical solution of partial differential equations many techniques have been suggested for integrating forwards in time, ranging from simple finite difference methods such as the Crank-Nicolson method to sophisticated finite element methods in time. In the present paper time discretizations introduced by Nbrsett' are evaluated with emphasis on simplicity of implementation in computer programs. The new methods are shown to be competitive with Crank-Nicolson.Our sole concern is time discretization. Assuming that a partial differential equation has been discretized in its space dimensions to give a system of ordinary differential equations, it is this system which we consider. The fact that such a separation of variables is performed rules out of consideration some well known difference schemes as the Dufort-Frankel method which is not derived in this way.To be specific, consider the diffusion convection equation defined for 0 < x < 1 and positive time t with initial and boundary conditions:

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“…Initial-boundary value problems of parabolic type serve as mathematical models in many practical applications [1], [5], [6], [10], [11], [13], [14].…”

Section: Introductionmentioning

confidence: 99%