Finite difference schemes satisfying an optimality condition for the unsteady heat equation

Domínguez-Mota, Francisco J.; Armenta, Sanzon Mendoza; Tinoco-Guerrero, Gerardo; Tinoco-Ruiz, José Gerardo

doi:10.1016/j.matcom.2014.02.007

Cited by 5 publications

(3 citation statements)

References 10 publications

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“…Nevertheless, in a similar way to the case of finite elements on very elongated triangles, convergence is expected to be achieved [17]. As follows from the numerical tests, the proposed heuristic scheme produces satisfactory results and, even though it might be in theory slightly less accurate than the scheme addressed in [8], it was as robust as the scheme defined by a local constrained optimization problem 4. In addition, in all the tests, it was notably faster due to the lack of constraints.…”

Section: Numerical Testmentioning

confidence: 68%

“…Problem (4) must be solved iteratively for the 3 × 3 structured subgrid around every inner grid node P i,j defined by G to provide the optimal local approximation to the differential operator (1). Very satisfactory results obtained with this scheme were reported in [8]. In this paper, we propose an alternative heuristic scheme based on an unconstrained optimization problem which is closely related to the constrained one.…”

Section: Finite Difference Schemes For Linear Operatorsmentioning

confidence: 99%

“…In [8], in order to get a second order scheme, we proposed to calculate the coefficients Γ 0 , Γ 1 , ..., Γ 8 by solving the constrained optimization problem…”

Section: Finite Difference Schemes For Linear Operatorsmentioning

confidence: 99%

See 2 more Smart Citations

An heuristic finite difference scheme on irregular plane regions

Domínguez-Mota¹,

Fernandez-Valdez²,

Ruiz-Diaz³

et al. 2014

ams

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In this paper, we present an heuristic finite difference scheme for the second-order linear operator, which is derived from an unconstrained least squares problem defined by the consistency condition on the residuals of order one, two and three in the Taylor expansion of the local truncation error. It is based on a non-iterative calculation of the difference coefficients and can be used to solve efficiently Poisson-like equations on non-rectangular domains which are approximated by structured convex grids.

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Section: Numerical Testmentioning

confidence: 68%

Section: Finite Difference Schemes For Linear Operatorsmentioning

confidence: 99%

See 1 more Smart Citation

An heuristic finite difference scheme on irregular plane regions

Domínguez-Mota¹,

Fernandez-Valdez²,

Ruiz-Diaz³

et al. 2014

ams

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Numerical solution of Richards’ equation of water flow by generalized finite differences

Chávez-Negrete

Domínguez-Mota

Santana-Quinteros

2018

Computers and Geotechnics

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Numerical solution of density-driven groundwater flows using a generalized finite difference method defined by an unweighted least-squares problem

Román-Gutiérrez

Chávez-Negrete

Domínguez-Mota

et al. 2022

Front. Appl. Math. Stat.

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Density-driven groundwater flows are described by nonlinear coupled differential equations. Due to its importance in engineering and earth science, several linearizations and semi-linearization schemes for approximating their solution have been proposed. Among the more efficient are the combinations of Newtonian iterations for the spatially discretized system obtained by either scalar homotopy methods, fictitious time methods, or meshless generalized finite difference method, with several implicit methods for the time integration. However, when these methods are used, some parameters need to be determined, in some cases, even manually. To overcome this problem, this paper presents a novel generalized finite differences scheme combined with an adaptive step-size method, which can be applied for solving the governing equations of interest on non-rectangular structured and unstructured grids. The proposed method is tested on the Henry and the Elder problems to verify the accuracy and the stability of the proposed numerical scheme.

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Finite difference schemes satisfying an optimality condition for the unsteady heat equation

Cited by 5 publications

References 10 publications

An heuristic finite difference scheme on irregular plane regions

An heuristic finite difference scheme on irregular plane regions

Numerical solution of Richards’ equation of water flow by generalized finite differences

Numerical solution of density-driven groundwater flows using a generalized finite difference method defined by an unweighted least-squares problem

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