On exact finite-difference schemes for hyperbolic and elliptic equations

Матус, П. П.; Irkhin, V. Yu.; Lapinska-Chrzczonowicz, Magdalena; Lemeshevsky, Sergey

doi:10.1134/s0012266107070130

Cited by 6 publications

(5 citation statements)

References 2 publications

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“…with the initial and boundary conditions (5). The difference scheme approximating the above problem is [8]…”

Section: Difference Schemes For a Semilinear Transport Equationmentioning

confidence: 99%

“…The authors have established that for the parabolic problems with travelling wave solutions the EDS may be constructed [7,8]. This paper refers to the travelling waves which arise in many problems such as heat transfer, combustion, reaction chemistry, fluid dynamics, plasma physics, soil-moisture, foam drainage, crystal growth, biological population genetics, cellular ecology, neurology and synergy [9,10].…”

Section: Introductionmentioning

confidence: 99%

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Exact difference schemes and schemes of higher order of approximation for convection-diffusion equation. I

Lapinska-Chrzczonowicz¹,

Матус²

2013

Annales UMCS, Informatica

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The initial-boundary value problem for a convection-diffusion equation ∂u ∂t + a ∂u ∂x = ∂ ∂x k(u) ∂u ∂x , (x, t) ∈ Q T , u(x, 0) = u 0 (x), 0 ≤ x ≤ l, u(0, t) = μ 1 (t), u(l, t) = μ 2 (t), 0 ≤ t ≤ T is considered. The difference scheme, approximating this problem, is constructed. It is shown that for traveling wave solutions the scheme is exact (EDS). The monotonicity of the scheme is also taken into consideration. Presented numerical experiments illustrate the theoretical results investigated in the paper.

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“…with the initial and boundary conditions (5). The difference scheme approximating the above problem is [8]…”

Section: Difference Schemes For a Semilinear Transport Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Exact difference schemes and schemes of higher order of approximation for convection-diffusion equation. I

Lapinska-Chrzczonowicz¹,

Матус²

2013

Annales UMCS, Informatica

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“…here h n i = x n i+1 − x n i is the space step at time t = t n . In [3,7] the EDS approximating problem (5) - (8) was constructed…”

Section: The Difference Scheme Of An Arbitrary Order Of Accuracymentioning

confidence: 99%

“…In [10] the investigations of the order of approximation, stability, and convergence of the high accuracy difference schemes for the nonlinear transfer equation ∂u ∂t + u ∂u ∂x = f (u) have been made. The EDS and the difference schemes of an arbitrary order of approximation for the parabolic equations with travelling wave solutions u(x, t) = U (x − at) were constructed in [7,8].…”

mentioning

confidence: 99%

Difference schemes of arbitrary order of accuracy for semilinear parabolic equations

Lapinska-Chrzczonowicz¹

2010

Annales UMCS, Informatica

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The Cauchy problem for a semilinear parabolic equation is considered. Under the conditions u(x, t) = X(x)T 1 (t) + T 2 (t), ∂u ∂x = 0, it is shown that the problem is equivalent to the system of two ordinary differential equations for which exact difference scheme (EDS) with special Steklov averaging and difference schemes with arbitrary order of accuracy (ADS) are constructed on the moving mesh. The special attention is paid to investigating approximation, stability and convergence of the ADS. The convergence of the iteration method is also considered. The presented numerical examples illustrate theoretical results investigated in the paper. IntroductionIn which cases an EDS or an ADS approximating nonlinear parabolic equation can be constructed? The paper deals with this question. The simple technique is presented and the main features of the constructed scheme are considered. Definition 2. [3]A difference scheme is exact if the truncation error is equal to zero or the exact solution agrees with the numerical solution at the grid nodes.The problem of constructing a difference scheme of high order of accuracy is topical. In papers [1,6] the EDSs and truncated difference schemes of an arbitrary rank were constructed for the nonlinear second order differential equation and for the systems of first-order nonlinear * lapkam@kul.pl Pobrane z czasopisma Annales AI-Informatica http://ai.annales.umcs.pl Data: 16/07/2020 03:37:39 U M C S 94 Difference schemes of arbitrary order of accuracy for semilinear parabolic equations τ m 2M +2 , where m, M are positive, natural constants.

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“…Exact finite-difference schemes for hyperbolic and nonlinear parabolic equations whose solutions have the form of a running wave with constant velocity, u(x, t) = Ψ(x − at), were constructed in [14][15][16] One of the directions of Matus' theoretical work is the investigation of well-posedness and blow up for IBVP for semilinear parabolic equations and numerical methods. In [11][12][13]17] he has obtained simple sufficient input data conditions, in which the solutions of differential and difference problems are globally bounded for all 0 t +∞.…”

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confidence: 99%