On the consistent two-side estimates for the solutions of quasilinear convection–diffusion equations and their approximations on non-uniform grids

Матус, П. П.; Poliakov, Dmitriy; Hieu, Le Minh

doi:10.1016/j.cam.2017.09.020

Cited by 5 publications

(1 citation statement)

References 4 publications

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“…On uniform grids [35], for the case of the one-dimensional convection-diffusion equation, there is the simplest approximations of first derivatives leading to upwind difference schemes. Such schemes are unconditionally monotone.…”

Section: Introductionmentioning

confidence: 99%

Monotone Finite-Difference Schemes With Second Order Approximation Based on Regularization Approach for the Dirichlet Boundary Problem of the Gamma Equation

2020

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We investigate the initial boundary value problem for the Gamma equation transformed from the nonlinear Black-Scholes equation for pricing option to a quasilinear parabolic equation of second derivative. Furthermore, two-side estimates for the exact solution are also provided. By using regularization principle, the unconditionally monotone second order approximation finite-difference scheme on uniform and nonuniform grids is generalized, in that the maximum principle is satisfied without depending on relations of the coefficients and grid parameters. By using the difference maximum principle, we acquired two-side estimates for difference solution for the arbitrary non-sign-constant input data. Finally, we also provide a proof for a priori estimate. It can be confirmed that the two-side estimates for difference solution are completely consistent with the differential problem. Otherwise, the maximal and minimal values of the difference solution is independent from the diffusion and convection coefficients.

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Section: Introductionmentioning

confidence: 99%

Monotone Finite-Difference Schemes With Second Order Approximation Based on Regularization Approach for the Dirichlet Boundary Problem of the Gamma Equation

2020

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On the Nonstandard Maximum Principle and Its Application for Construction of Monotone Finite-Difference Schemes for Multidimensional Quasilinear Parabolic Equations

Hieu,

Xuan,

Thanh

2024

Ukr Math J

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Second Order Monotone Difference Schemes with Approximation on Non-Uniform Grids for Two-Dimensional Quasilinear Parabolic Convection-Diffusion Equations

Hieu

Thanh

Prasath

2020

Vestnik St.Petersb. Univ.Math.

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Настоящая работа посвящена построению монотонных разностных схем второго порядка локальной аппроксимации на неравномерных сетках по пространству для двумерного квазилинейного параболического уравнения конвекции-диффузии. С помощью разностного принципа максимума устанавливаются двусторонние оценки разностного решения и доказывается важная априорная оценка в равномерной норме C. Интересно отметить, что максимальное и минимальное значения разностного решения не зависят от коэффициентов диффузии и конвекции.Ключевые слова: неравномерная сетка, принцип максимума, принцип регуляризации, монотонная разностная схема, уравнение конвекции-диффузии.

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On the consistent two-side estimates for the solutions of quasilinear convection–diffusion equations and their approximations on non-uniform grids

Cited by 5 publications

References 4 publications

Monotone Finite-Difference Schemes With Second Order Approximation Based on Regularization Approach for the Dirichlet Boundary Problem of the Gamma Equation

Monotone Finite-Difference Schemes With Second Order Approximation Based on Regularization Approach for the Dirichlet Boundary Problem of the Gamma Equation

On the Nonstandard Maximum Principle and Its Application for Construction of Monotone Finite-Difference Schemes for Multidimensional Quasilinear Parabolic Equations

Second Order Monotone Difference Schemes with Approximation on Non-Uniform Grids for Two-Dimensional Quasilinear Parabolic Convection-Diffusion Equations

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