sci2区top微分求积法一种快速解的技术非线性偏微分方程
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i=l
i =
1, 2,..,, N.
(3)
There are many ways of determining the coefficients uii . One method for determining these coefficients will be discussed below. Substitution of Eq. (3) into Eq. (1) yields the set of N ordinary differential equations
Departments of Mathematics, Electrical Engineering and Medicine,
B. G. KASHEF
Department of Electrical Engineering, University of Southern California Los Angeles, California 90007
1. INTRODUCTION
The numerical solution of nonlinear partial differential equations plays a prominent role in numerical weather forecasting [l], optimal control theory [2], radiative transfer [3], and many other areas of physics, engineering, and biology. In many casesall that is desired is a moderately accurate solution at a few points which can be calculated rapidly. The standard finite difference methods currently
3. DETERMINATION OF WEIGHTING COEFFICIENTS
In order to determine appropriate coefficients in the approximation U&i) s 5 Ui&j),
j=l
i =
1, 2,..., N,
(1)
we can proceed by analogy with the classical quadrature case, demanding that Eq. (1) be exact for all polynomials of degreeless than or equal to N - 1. The test function p&) = &l, k = l,..., N, for arbitrary distinct xi leads to the set of linear algebraic equations
with initial conditions
40, Xi> = w4,
i=l,2
,***,N.
(5)
42
BELLMAN, KASHEF, AND CAST1
Hence, under the assumption that (3) is valid, we have succeededin reducing the task of solving Eq. (1) to the task of solving a set of ordinary differential equations with prescribed initial values. The numerical solution of such a system, Eq. (5), is a simple task for modern electronic computers using standard programs with minimal computing time and storage (in contrast to the case of partial differential equations). The additional storage neededfor the solution of a partial differential equation using the differential quadrature technique is that of storing the N x N matrix aii . Considering that in practice it turns out that relatively low order differential quadrature is all that is needed,the total amount of storage and time required on the machine is thus quite low. We also note that the number of arithmetic operations to be performed for every point is the N additions and multiplications, Eq. (3), plus the amount needed for the solution of the set of ordinary differential equations (which is, of course, method dependent).
* Supported by the National Science Foundation under Grant No. GP 29049 and the Atomic Energy Commission under Grant No. AT(4O-3), 113, Project No. 19.
AND
J. CASTI
Systems Control, Inc. Palo Alto, California 94306 Received May 13, 1971
The numerical solution of nonlinear partial differential equations plays a prominent role in numerical weather forecasting, optimal control theory, radiative transfer, and many other areas of physics, engineering, and biology. In many cases all that is desired is a moderately accurate solution at a few points which can be calculated rapidly. In this paper we wish to present a simple direct technique whichBaidu Nhomakorabeacan be applied in a large number of cases to circumvent the difficulties of programming complex algorithms for the computer, as well as excessive use of storage and computer time. We illustrate this technique with the solution of some partial differential equations arising in various simplified models of fluid flow and turbulence.
40
Copyright All rights 0 1972 by Academic Press, Inc. of reproduction in any form reserved.
DIFFERENTIAL QUADRATURE
41
in use have the characteristic that the solution must be calculated at a large number of points in order to obtain moderately accurate results at the points of interest. Consequently, both the computing time and storage required often prohibit the calculation. Furthermore, the mathematical techniques involved in finite difference schemesor in the Fourier transform methods, are often quite sophiscated and thus not easily learned or used. In this paper we wish to present a technique which can be applied in a large number of cases to circumvent both the above difficulties. We illustrate this technique with the solution of some partial differential equations arising in various simplified models of fluid flow and turbulence. It is not easy to provide any stability analysis becauseof the nonlinearity of the equations considered. In subsequentpapers, however, we will examine the stability of the numerical schemeapplied to various linear equations since some questions of independent interest arise. We shall also discuss a number of other types of boundary conditions and quadratures. 2. DIFFERENTIAL QUADRATURE Consider the nonlinear first-order partial differential equation Ut = g(t, x, u, u& -Go < x < co, t > 0 with initial condition 64x) = w,
JOURNAL
OF COMPUTATIONAL
PHYSICS
10,
40-52 (1972)
Differential Quadrature: A Technique for the Rapid Solution of Nonlinear Partial Differential Equations*
RICHARD BELLMAN
(2)
(1)
an equation arising in many mathematical models of physical processes.Let us make the assumption that the function u satisfying Eqs. (1) and (2) is sufficiently smooth to allow us to write the approximate relation Ux(tyXi) 5% $J QijU(tyXj),
i =
1, 2,..,, N.
(3)
There are many ways of determining the coefficients uii . One method for determining these coefficients will be discussed below. Substitution of Eq. (3) into Eq. (1) yields the set of N ordinary differential equations
Departments of Mathematics, Electrical Engineering and Medicine,
B. G. KASHEF
Department of Electrical Engineering, University of Southern California Los Angeles, California 90007
1. INTRODUCTION
The numerical solution of nonlinear partial differential equations plays a prominent role in numerical weather forecasting [l], optimal control theory [2], radiative transfer [3], and many other areas of physics, engineering, and biology. In many casesall that is desired is a moderately accurate solution at a few points which can be calculated rapidly. The standard finite difference methods currently
3. DETERMINATION OF WEIGHTING COEFFICIENTS
In order to determine appropriate coefficients in the approximation U&i) s 5 Ui&j),
j=l
i =
1, 2,..., N,
(1)
we can proceed by analogy with the classical quadrature case, demanding that Eq. (1) be exact for all polynomials of degreeless than or equal to N - 1. The test function p&) = &l, k = l,..., N, for arbitrary distinct xi leads to the set of linear algebraic equations
with initial conditions
40, Xi> = w4,
i=l,2
,***,N.
(5)
42
BELLMAN, KASHEF, AND CAST1
Hence, under the assumption that (3) is valid, we have succeededin reducing the task of solving Eq. (1) to the task of solving a set of ordinary differential equations with prescribed initial values. The numerical solution of such a system, Eq. (5), is a simple task for modern electronic computers using standard programs with minimal computing time and storage (in contrast to the case of partial differential equations). The additional storage neededfor the solution of a partial differential equation using the differential quadrature technique is that of storing the N x N matrix aii . Considering that in practice it turns out that relatively low order differential quadrature is all that is needed,the total amount of storage and time required on the machine is thus quite low. We also note that the number of arithmetic operations to be performed for every point is the N additions and multiplications, Eq. (3), plus the amount needed for the solution of the set of ordinary differential equations (which is, of course, method dependent).
* Supported by the National Science Foundation under Grant No. GP 29049 and the Atomic Energy Commission under Grant No. AT(4O-3), 113, Project No. 19.
AND
J. CASTI
Systems Control, Inc. Palo Alto, California 94306 Received May 13, 1971
The numerical solution of nonlinear partial differential equations plays a prominent role in numerical weather forecasting, optimal control theory, radiative transfer, and many other areas of physics, engineering, and biology. In many cases all that is desired is a moderately accurate solution at a few points which can be calculated rapidly. In this paper we wish to present a simple direct technique whichBaidu Nhomakorabeacan be applied in a large number of cases to circumvent the difficulties of programming complex algorithms for the computer, as well as excessive use of storage and computer time. We illustrate this technique with the solution of some partial differential equations arising in various simplified models of fluid flow and turbulence.
40
Copyright All rights 0 1972 by Academic Press, Inc. of reproduction in any form reserved.
DIFFERENTIAL QUADRATURE
41
in use have the characteristic that the solution must be calculated at a large number of points in order to obtain moderately accurate results at the points of interest. Consequently, both the computing time and storage required often prohibit the calculation. Furthermore, the mathematical techniques involved in finite difference schemesor in the Fourier transform methods, are often quite sophiscated and thus not easily learned or used. In this paper we wish to present a technique which can be applied in a large number of cases to circumvent both the above difficulties. We illustrate this technique with the solution of some partial differential equations arising in various simplified models of fluid flow and turbulence. It is not easy to provide any stability analysis becauseof the nonlinearity of the equations considered. In subsequentpapers, however, we will examine the stability of the numerical schemeapplied to various linear equations since some questions of independent interest arise. We shall also discuss a number of other types of boundary conditions and quadratures. 2. DIFFERENTIAL QUADRATURE Consider the nonlinear first-order partial differential equation Ut = g(t, x, u, u& -Go < x < co, t > 0 with initial condition 64x) = w,
JOURNAL
OF COMPUTATIONAL
PHYSICS
10,
40-52 (1972)
Differential Quadrature: A Technique for the Rapid Solution of Nonlinear Partial Differential Equations*
RICHARD BELLMAN
(2)
(1)
an equation arising in many mathematical models of physical processes.Let us make the assumption that the function u satisfying Eqs. (1) and (2) is sufficiently smooth to allow us to write the approximate relation Ux(tyXi) 5% $J QijU(tyXj),