Fundamental Concepts in the Numerical Solution of Differential Equations

Botha, J. F.; Pinder, George F.; Patera, Anthony T.

doi:10.1115/1.3167711

Cited by 12 publications

(22 citation statements)

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“…HYBRID ALGORITHMS FOR PARTIAL DIFFERENTIAL EQUATIONS Similar ideas as described in Section II can be applied to solve partial differential equations. This section proposes two hybrid algorithms combining evolutionary computation techniques with the difference method [2] for solving the Dirichlet problem [2].…”

Section: B Numerical Experimentsmentioning

confidence: 99%

“…Built on the work of solving linear equations, this paper also proposes two hybrid algorithms for solving partial differential equations, i.e., the Dirichlet problem [2]. Both hybrid algorithms use the difference method [2] to discretise the partial differential equations into a linear system first and then solved it.…”

Section: Introductionmentioning

confidence: 99%

“…Both hybrid algorithms use the difference method [2] to discretise the partial differential equations into a linear system first and then solved it.…”

Section: Introductionmentioning

confidence: 99%

“…Then these individuals can exchange information through crossover. Experimental results show that the hybrid algorithm can solve equations within a small fraction of time needed by the classical SOR method for a number of problems we have tested.Built on the work of solving linear equations, this paper also proposes two hybrid algorithms for solving partial differential equations, i.e., the Dirichlet problem [2]. Both hybrid algorithms use the difference method [2] to discretise the partial differential equations into a linear system first and then solved it.…”

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confidence: 99%

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Solving equations by hybrid evolutionary computation techniques

Jiang

Yao³

2000

IEEE Trans. Evol. Computat.

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Evolutionary computation techniques have mostly been used to solve various optimisation and learning problems. This paper describes a novel application of evolutionary computation techniques to equation solving. Several combinations of evolutionary computation techniques and classical numerical methods are proposed to solve linear and partial differential equations. The hybrid algorithms have been compared with the well-known classical numerical methods. The experimental results show that the proposed hybrid algorithms outperform the classical numerical methods significantly in terms of effectiveness and efficiency. Index TermsLinear equations, Partial differential equations, Adaptation, Hybrid algorithms. I. INTRODUCTIONThere has been a huge increase in the number of papers and successful applications of evolutionary computation techniques in a wide range of areas in recent years. Almost all these applications can be classified as evolutionary optimisation (either numerical or combinatorial) or evolutionary learning (supervised, reinforcement or unsupervised). This paper presents a very different and novel application of evolutionary computation techniques in equation solving, i.e., solving linear and partial differential equations by simulated evolution.One of the best-known numerical methods for solving linear equations is the successive over relaxation (SOR) method [1]. However, it is often very difficult to estimate the optimal relaxation factor, which is a key parameter of the SOR method. This paper proposes a hybrid algorithm combining the SOR method with evolutionary computation techniques. The hybrid algorithm does not require a user to guess or estimate the optimal relaxation factor. The algorithm "evolves" it.Unlike most other hybrid algorithms where an evolutionary algorithm is used a wrapper around another algorithm (often a classical algorithm), the hybrid algorithm proposed in this paper integrates the SOR method with evolutionary computation techniques, such as crossover and mutation. It makes better use of a population by employing different equation solving strategies for different individuals in the population. Then these individuals can exchange information through crossover. Experimental results show that the hybrid algorithm can solve equations within a small fraction of time needed by the classical SOR method for a number of problems we have tested.Built on the work of solving linear equations, this paper also proposes two hybrid algorithms for solving partial differential equations, i.e., the Dirichlet problem [2]. Both hybrid algorithms use the difference method [2] to discretise the partial differential equations into a linear system first and then solved it.Fogel and Atmar [3] used linear equation solving as test problems for comparing crossover and inversion operators and Gaussian mutation in an evolutionary algorithm. A linear system of the form

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Section: B Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Both hybrid algorithms use the difference method [2] to discretise the partial differential equations into a linear system first and then solved it.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Solving equations by hybrid evolutionary computation techniques

Jiang

Yao³

2000

IEEE Trans. Evol. Computat.

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“…The use of collocation methods for the solution of partial differential equations (PDE) has become a subject of intensive interest in the past [Lanczos 1938;Ronto 1971;Russell and Shampine 1972;Finlayson 1972;De Boor and Swartz 1973;Diaz 1977;Houstis 1978;Botha and Pinder 1983]. While most research has focused on the local interpolation of a variable, not many works have appeared concerning the global collocation in two-dimensional problems [Frind and Pinder 1979;Hayes 1980;Van Blerk and Botha 1993].…”

Section: Introductionmentioning

confidence: 99%

Untitled

2008

JOMMS

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See inside back cover or http://www.jomms.org for submission guidelines.Regular subscription rate: $500 a year. We study an axially symmetric frictionless contact problem between a nonhomogeneous elastic halfspace and a rigid base that has a small axisymmetric surface recess. We reduce the problem to solving Fredholm integral equations, solve these equations numerically, and establish a relationship between the applied pressure and the gap radius. We find and graph the effects of nonhomogeneity on the normal pressure, critical pressure and on the surface displacement.

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Single‐degree of freedom Hermite collocation for multiphase flow and transport in porous media

Fedele

McKay

Pinder

et al. 2004

Numerical Methods in Fluids

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SUMMARYThe classical collocation method using Hermite polynomials is computationally expensive as the dimensionality of the problem increases. Because of the use of a C 1 -continuous basis, the method generates two, four and eight unknowns per node for one, two and three-dimensional problems, respectively. In this paper we propose a numerical strategy to reduce the nodal unknowns to a single degree of freedom at each node. The reduction of the unknowns is due to the use of Lagrangian polynomials to approximate the ÿrst-order derivatives over the minimal compact stencil surrounding each node. For the solvability of the problem the reduction of the number of collocation equations is done by a nodal weighting strategy. We have applied the proposed approach to enhance the e ciency of a collocationbased multiphase ow and transport simulator. Benchmark cases illustrate the higher performance of the new methodology when compared to classical Hermite collocation.

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Fundamental Concepts in the Numerical Solution of Differential Equations

Cited by 12 publications

References 0 publications

Solving equations by hybrid evolutionary computation techniques

Solving equations by hybrid evolutionary computation techniques

Untitled

Single‐degree of freedom Hermite collocation for multiphase flow and transport in porous media

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