<p>In this paper, the first initial-boundary value problem for a non-stationary loaded differential equation of convection-diffusion in a multidimensional domain is studied in a rectangular domain. Two difference schemes are proposed that approximate the original multidimensional problem with the order of accuracy <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varvec{O(|h|^2+\tau ^2)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">O</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mo mathvariant="bold" stretchy="false">|</mo> <mi mathvariant="bold-italic">h</mi> <msup> <mo mathvariant="bold" stretchy="false">|</mo> <mn mathvariant="bold">2</mn> </msup> <mo mathvariant="bold">+</mo> <msup> <mi mathvariant="bold-italic">τ</mi> <mn mathvariant="bold">2</mn> </msup> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varvec{O(|h|^4+\tau ^2)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">O</mi> <mo mathvariant="bold" stretchy="false">(</mo> <mo mathvariant="bold" stretchy="false">|</mo> <mi mathvariant="bold-italic">h</mi> <msup> <mo mathvariant="bold" stretchy="false">|</mo> <mn mathvariant="bold">4</mn> </msup> <mo mathvariant="bold">+</mo> <msup> <mi mathvariant="bold-italic">τ</mi> <mn mathvariant="bold">2</mn> </msup> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Using the method of energy inequalities, a priori estimates for solutions of the proposed difference problems are obtained. These estimates allow us to establish the uniqueness and stability of solutions depending on the specified conditions. For each difference scheme, it is proved that the numerical solution approaches the solution of the original differential problem if the latter belongs to the class of sufficiently smooth functions.</p>

DIFFERENCE SCHEMES FOR THE LOADED CONVECTION-DIFFUSION EQUATION IN A MULTIDIMENSIONAL DOMAIN

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发表日期：

2026年2月5日

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Murat KH. Beshtokov,
Zaryana V. Beshtokova

摘要

In this paper, the first initial-boundary value problem for a non-stationary loaded differential equation of convection-diffusion in a multidimensional domain is studied in a rectangular domain. Two difference schemes are proposed that approximate the original multidimensional problem with the order of accuracy $\varvec{O(|h|^2+\tau ^2)}$ $O (| h |^{2} + τ^{2})$ and $\varvec{O(|h|^4+\tau ^2)}$ $O (| h |^{4} + τ^{2})$ . Using the method of energy inequalities, a priori estimates for solutions of the proposed difference problems are obtained. These estimates allow us to establish the uniqueness and stability of solutions depending on the specified conditions. For each difference scheme, it is proved that the numerical solution approaches the solution of the original differential problem if the latter belongs to the class of sufficiently smooth functions.