Numerical Solution of 2-D Burgers’ Equations Using RBF-Based Mesh-free Method

An important feature of traditional numerical methods in computational fluid dynamic problems is that a corresponding mesh is required as a computational framework to solve the governing partial differential equation. These methods encounter many difficulties in handling some special problems with deformable boundaries, free surface, moving material interface and so on. Mesh-free method attracts a great deal of interest in recent years due to their lack of dependency on a mesh or grid in the domain under consideration. It is quite important to note that the mesh-free methods and the traditional numerical methods are different with regard to definition of shape functions. While in traditional numerical approaches, shape functions are constructed with due attention on element shape, in mesh-free method, shape functions are constructed for a particular point under consideration. One of the most important issues in mesh-free methods is to construct an accurate and stable interpolating function. In this paper, a truly mesh-free numerical technique is developed to solve the system of two-dimensional Burgers' equations using radial basis interpolation function in order to construct mesh-free shape functions. The developed RBF-based mesh-free method considers the partial differential equations in strong form. The strong form of the PDE is solved using the developed mesh-free model and the solutions are compared to the analytical solution and also the solutions obtained from an implicit finite difference approach implemented earlier. The proposed numerical scheme is found to be robust and stable. Compared to the exact solution of system of two-dimensional Burgers’ equations, it is demonstrated that the developed mesh-free technique results in satisfactory solutions for irregular boundary domain and various values of Reynolds number up to 500.

کلیدواژه ها:

Numerical computations ، Mesh-free methods ، RBF interpolation technique ، Burgers' equation ، strong form of PDEs