O(h⁵_k) accurate finite difference method for the numerical solution of fourth order two point boundary value problems on geometric meshes

• Autorzy: Jha N. , Bieniasz L. K.

Warianty tytułu

PL

Metoda różnicowa o dokładności O(h⁵_k), do rozwiązywania dwupunktowych zagadnień brzegowych czwartego rzędu na siatkach geometrycznych

• Języki publikacji: EN

Abstrakty

EN

Two point boundary value problems for fourth order, nonlinear, singular and non-singular ordinary differential equations occur in various areas of science and technology. A compact, three point finite difference scheme for solving such problems on nonuniform geometric meshes is presented. The scheme achieves a fifth or sixth order of accuracy on geometric and uniform meshes, respectively. The proposed scheme describes the generalization of Numerov-type method of Chawla (IMA J Appl Math 24:35-42, 1979) developed for second order differential equations. The convergence of the scheme is proven using the mean value theorem, irreducibility, and monotone property of the block tridiagonal matrix arising for the scheme. Numerical tests confirm the accuracy, and demonstrate the reliability and efficiency of the scheme. Geometric meshes prove superior to uniform meshes, in the presence of boundary and interior layers.

PL

Dwupunktowe zagadnienia z warunkami brzegowymi, dla nieliniowych, osobliwych i nieosobliwych równań różniczkowych zwyczajnych czwartego rzędu, występują w różnych obszarach nauki i techniki. Zaprezentowano kompaktowy, trzypunktowy schemat różnicowy do rozwiązywania takich problemów na niejednorodnych siatkach geometrycznych. Schemat ten osiąga dokładność piątego lub szóstego rzędu, odpowiednio na siatkach geometrycznych lub jednorodnych. Proponowany schemat przedstawia uogólnienie metody typu Numerowa, autorstwa Chawli (IMA J Appl Math 24:35-42, 1979), opracowanej dla równań różniczkowych drugiego rzędu. Udowodniono zbieżność schematu, korzystając z twierdzenia o własności średniej, nieredukowalności oraz monotoniczności macierzy blokowo-trójdiagonalnej wynikającej ze schematu. Testy numeryczne potwierdzają dokładność, oraz demonstrują niezawodność i wydajność schematu. Siatki geometryczne wykazują przewagę nad siatkami jednorodnymi, w obecności warstw brzegowych i wewnętrznych.

Słowa kluczowe

EN

geometric mesh; finite difference method; compact scheme; singularity; stiff equations; Korteweg–de Vries equation; maximum absolute errors

PL

siatka geometryczna; metoda różnic skończonych; schemat kompaktowy; osobliwość; równania sztywne; równanie Kortewega-de Vries; maksymalne błędy bezwzględne

• Wydawca: Wydawnictwo Politechniki Krakowskiej im. Tadeusza Kościuszki
• Czasopismo: Czasopismo Techniczne. Nauki Podstawowe
• Rocznik: 2016
• Tom: Y. 113, iss. 1-NP
• Strony: 55--72
• Opis fizyczny: Bibliogr. 43 poz., wz., tab.

Twórcy

autor

Jha N.

• Department of Mathematics, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi, India

autor

Bieniasz L. K.

• Institute of Network Computing, Faculty of Physics, Mathematics and Computer Science, Cracow University of Technology

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