Numerical study of non-linear waves for one-dimensional planar, cylindrical and spherical flow using B-spline finite element method

Azhar Iqbal; Abdullah M. Alsharif; Sahar Albosaily; Azhar Iqbal; Abdullah M. Alsharif; Sahar Albosaily

doi:10.3934/math.2022844

AIMS Mathematics

2022, Volume 7, Issue 8: 15417-15435. doi: 10.3934/math.2022844

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Numerical study of non-linear waves for one-dimensional planar, cylindrical and spherical flow using B-spline finite element method

1.
Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar 31952, Kingdom of Saudi Arabia
2.
Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
3.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia

Received: 20 May 2022 Revised: 08 June 2022 Accepted: 14 June 2022 Published: 10 June 2022
MSC : 58B05, 55U40, 34A34, 35C10, 34A45

In a recent study, an evolution equation is found for waves' behavior at far-field with relaxation mode of molecules. An analytical technique was used to solve this evolution problem, which is a generalized Burger equation. The analytical approach has limitations and requires a very accurate initial guess by a trial method. In this paper, the evolution equation for one-dimensional planar, cylindrical, and spherical flow in the presence of relaxation mode is solved using a collocation approach with a cubic B-spline function. The numerical results are graphed and compared with the exact solution for planar flow. The obtained numerical results match the exact solution quite well and show that the technique is quite reliable and can deal with the nonlinearity involved in the present problem. Results have also been obtained for cylindrical and spherical flow at the far-field. The obtained numerical results show that the present approach with the cubic B-spline function works well and accurately. Fourier stability analysis is used to investigate the stability of the cubic B-spline collocation method.
- far-field,
- conservation laws,
- stability analysis,
- partial differential equation,
- B-spline function,
- finite element method,
- relaxation mode
Citation: Azhar Iqbal, Abdullah M. Alsharif, Sahar Albosaily. Numerical study of non-linear waves for one-dimensional planar, cylindrical and spherical flow using B-spline finite element method[J]. AIMS Mathematics, 2022, 7(8): 15417-15435. doi: 10.3934/math.2022844

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Abstract

In a recent study, an evolution equation is found for waves' behavior at far-field with relaxation mode of molecules. An analytical technique was used to solve this evolution problem, which is a generalized Burger equation. The analytical approach has limitations and requires a very accurate initial guess by a trial method. In this paper, the evolution equation for one-dimensional planar, cylindrical, and spherical flow in the presence of relaxation mode is solved using a collocation approach with a cubic B-spline function. The numerical results are graphed and compared with the exact solution for planar flow. The obtained numerical results match the exact solution quite well and show that the technique is quite reliable and can deal with the nonlinearity involved in the present problem. Results have also been obtained for cylindrical and spherical flow at the far-field. The obtained numerical results show that the present approach with the cubic B-spline function works well and accurately. Fourier stability analysis is used to investigate the stability of the cubic B-spline collocation method.

References

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