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

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.

Keywords

• far-field
• conservation laws
• stability analysis
• partial differential equation
• b-spline function
• finite element method
• relaxation mode