FINITE ELEMENT METHOD SCHEMES OF HIGHER ACCURACY FOR SOLVING NON-STATIONARY FOURTH-ORDER EQUATIONS
DOI:
https://doi.org/10.26577/JMMCS.2023.v118.i2.05Keywords:
finite element method, difference schemes, stability, convergence, accuracyAbstract
High-order Sobolev-type equations are mathematical models used in many applied problems. As is known, in many cases, it is difficult to obtain analytical solutions to high-order equations; therefore, they are mainly solved by numerical methods. At present, the method of straight lines is often used to solve non-stationary problems of mathematical physics; in this method, discretization is first realized only in spatial variables, and the resulting system of ordinary differential equations of high dimension is solved by finite difference methods or finite elements of higher accuracy. In this study, for a system of ordinary differential equations of the fourth order, new multi-parameter difference schemes of higher accuracy based on the finite element method are constructed. The presence of parameters in the scheme makes it possible to regularize the schemes in order to optimize the implementation algorithm and the accuracy of the scheme. The stability and convergence of the constructed difference schemes are also proved, and accuracy estimates are obtained on their basis. An algorithm for the implementation of the constructed difference schemes is presented. The results obtained can be further applied in the numerical solution to initial-boundary value problems for the equations of dynamics of a compressible stratified rotating fluid, magnetic gas dynamics, ion-acoustic waves in a magnetized plasma, spin waves in magnets, cold plasma in an external magnetic field, etc.
References
[2] Moskalkov M.N., Utebaev D. Convergence of Centered Difference Schemes for a System of Two–Dimensional Equations of Acoustics, Journal of Mathematical Sciences, Vol. 58, No. 3, (1992): 229–235. DOI: 10.1007/BF01098331.
[3] Utebaev D. Ob odnom metode chislennogo resheniia operatornogo differentsialnogo uravneniia vtorogo poriadka [On a Method for the Numerical Solution of a Second-Order Operator Differential Equation], (DAN RUz., Ser. matematika, tekhnicheskie nauki, estestvoznanie, No 1, 2007): 31-34.
[4] Moskalkov M.N., Utebaev D. Comparison of Some Methods for Solving the Internal Wave Propagation Problem in a Weakly Stratified Fluid, Mathematical Models and Computer Simulations, Vol. 3, No. 2, (2011): 264–271. DOI: 10.1134/S2070048211020086.
[5] Moskalkov M.N., Utebaev D. Convergence of the Finite Element Scheme for the Equation of Internal Waves, Cybernetics and Systems Analysis, Vol. 47, No. 3, (2011): 459 – 465.
[6] Moskalkov M.N., Utebaev D. Finite Element Method for the Gravity-Gyroscopic Wave Equation, Zhurnal obchisliuvalno ̈ı ta prikladno ̈ı matematiki, Vol. 101, No 2, (2010): 97-104.
[7] Kolkovska N., Angelow K. A Multicomponent Alternating Direction Method for Numerical Solution of Boussinesq Paradigm Equation, International Conference on Numerical Analysis and Its Applications, (2013): 371-378. DOI: 10.1007/978-3-642-41515-9−41.
[8] Hussain K.A., Ismail F., Senu N. Direct Numerical Method for Solving a Class of Fourth-order Partial Differential Equation, Global Journal of Pure and Applied Mathematics, Vol. 12, No. 2. (2016): 1257-1272. http://ripublication.com/Volume/gjpamv12n2.htm
[9] Hussain K.A., Ismail F., Senu N. Solving Directly Special Fourth-order Ordinary Differential Equations Uzing Runge-Kutta Type Method, Journal of Computational and Applied Mathematics, Vol. 306, (2013): 179-199. DOI: 10.1016/j.cam.2016.04.002.
[10] Taiwo O.A., Ogunlaran O.M. Numerical Soiution of Fourth-order Ordinary Differential Equations by Cubic Spline Collocation Tau Method, Journal of Mathematics and Statistics, Vol. 4, No. 4, (2008): 264-268. DOI: 10.1080/00207169708804551.
[11] Talwar J., Mohanty R. K. A Class of Numerical Methods for the Solution of Fourth-order Ordinary Differential Equations in Polar Coordinates, Advances in Numerical Analysis, Vol. 2012, Article ID 626419, (2012): 1-20. DOI: 10.1155/2012/626419.
[12] Singh D. I., Singh G. Numerical Study for Solving Fourth-order Ordinary Differential Equations, International Journal for Research in Engineering Application and Management, Vol. 05, No. 01, (2019): 638-642. DOI: 10.18231/2454- 9150.2019.0371.
[13] Kholodov S.E. Volnovye dvizheniia v szhimaemoi stratifitsirovannoi vrashchaiushcheisia zhidkosti [Wave Motions in a Compressible Stratified Rotating Fluid], Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki, T. 47, No 12, (2007): 2101-2109.
[14] Uizem Dzh. Lineinye i nelineinye volny [Linear and Non-Linear Waves], (M.: Mir, 1977): 622.
[15] Sveshnikov A.G., Alshin A.B., Korpusov M.O., Pletner Iu.D. Lineinye i nelineinye uravneniia sobolevskogo tipa [Linear and Non-Linear Equations of the Sobolev Type], (M.: FIZMATLIT, 2007): 736.
[16] Samarskii A.A. The Theory of Difference Schemes, (Vol. 240 of Pure and Applied Mathematics, Marcel Dekker Inc., New York, Basel, 2001): 786.
[17] Samarskii A. A., Gulin A. V. Ustoichivost’ raznostnykh skhem [The Stability of Difference Schemes], (M. Nauka, 1973): 415.
