Spectral Differentiation and Mimetic Methods for Solving the Scalar Burger’s Equation

Authors

  • Bertha K. Rodriguez-Chavez Escuela de Matemáticas, Universidad Nacional de Trujillo, Trujillo, Perú.
  • Yessica E. Zarate-Pedrera Departamento de Matemáticas, Universidad Nacional de Trujillo, Trujillo, Perú.

DOI:

https://doi.org/10.17268/sel.mat.2024.02.05

Keywords:

Burger’s equation, spectral differentiation, mimetic method

Abstract

In the present work, the spectral differentiation method was studied to solve the scalar  Burger’s partial differential equation. This equation has been of considerable physical interest as it can be regarded as a simplified version of the Navier-Stokes equations. Through this study, the spectral differentiation method and its convergence were described; additionally, the mimetic method and the use of the MOLE library for numerically solving the scalar Burger’s equation were presented.

References

Landajuela M. Burgers equation. BCAM Internship report: Basque Center for Applied Mathematics; 2011.

Quarteroni A, Sacco R, Saleri F. Numerical Mathematics. Springer; 2010.

Basdevant C, Deville M, Haldenwang P, Lacroix JM, Ouazzani J, Peyret R, Orlandi P, Patera AT. Spectral and finite difference solutions of the Burgers equation. Computer & fluids, 1986 14(1):23-41.

Gottlieb D, Orszag SA. Numerical Analysis of Spectral Methods: Theory and Applications. SIAM; 1977.

Reza M. Numerical solution of the Burgers’ equation using Fourier spectral method and comparison with finite difference method. 7th International conference on computational heat and mass transfer, 2011; 27(4).

Cía A. Métodos pseudoespectrales para la resolución numérica de ecuaciones en derivadas parciales[tesis maestría]. Respositorio Universidad de Valladolid, Facultad de ciencias; 2020.

Bernardi C, Maday Y. Handbook of numerical analysis. Techniques of Scientific Computing. 1997; 5(2):209-485.

Folland GB. Fourier analysis and its applications. Brooks/Cole Publishing Company. 1992.

Schlatter P. Spectral methods. Lecture notes KTH. 2009.

Bochev PB, Hyman JM. Principles of Mimetic Discretizations of Differential Operators. In: Arnold DN, Bochev PB, Lehoucq RB, Nicolaides RA, Shashkov M. (eds) Compatible Spatial Discretizations. The IMAVolumes in Mathematics and its Applications, vol 142. Springer, New York. 2006.

Adams RA, Fournier JJF. Sobolev Spaces. Elsevier Science. 2003; pp 59- 79.

Canuto C, Quarteroni A, Hussaini MY, Zang TA. Spectral Methods, fundamentals in single domains. Springer Scientific computation. 2006; 27(4):118-166.

Canuto C, Quarteroni A, Hussaini MY, Zang TA. Spectral methods in fluid dynamics. Springer series in computational physics. 1988; 36(3):31-65, 275-280.

Castillo J, Grone R. A matrix analysis approach to higher-order approximations for divergence and gradients satisfying a global conservation law. SIAM Journal. 2003; 25(1):128-142.

Corbino J, Castillo JE. High Order Mimetic Finite Difference Operators Satisfying a Gauss Divergence Theorem. J Appl Computat Math. 2018; 7: 387. doi: 10.4172/2168-9679.1000387

Andreucci D. et al. Some Numerical Results on Chemotactic Phenomena in Stem Cell Therapy for Cardiac Regeneration. Mathematics, 2024; 12(13), 1937. https://doi.org/10.3390/ math12131937

Published

2024-12-28

How to Cite

Rodriguez-Chavez, B. K., & Zarate-Pedrera, Y. E. (2024). Spectral Differentiation and Mimetic Methods for Solving the Scalar Burger’s Equation. Selecciones Matemáticas, 11(02), 259 - 270. https://doi.org/10.17268/sel.mat.2024.02.05