Numerical solution of the advection and Burgers equations using the point collocation-based method of finite spheres (PCMFS)

Abstract

The numerical solution of the inviscid advection problem and of the Burgers equation in the one and two-dimensional cases is studied, using the point collocation-based method of finite spheres (PCMFS [1]). The PCMFS is a meshfree numerical technique developed for the solution of partial differential equations on complex domains in which spatial discretization is performed on a scattered distribution of nodal points using the moving least squares (MLS) technique. The point collocation method [2] is used as the weighted residual scheme. The burden of mesh generation and remeshing, that complicates the numerical solution of most fluid dynamic problems using traditional finite element or finite volume approaches, is therefore mitigated. Each MLS shape function is compactly supported on a ball of radius r. Temporal discretization is performed using a first-order backward difference scheme. The solution accuracy is studied in a three-parameter space including the spacing between successive nodes ∆x, the temporal discretization parameter ∆t and the relative radius ρ=r/∆x. For both equations the same tendencies for the best choice of parameters is observed. Reducing the spatial discretization parameter ∆x minimizes errors in terms of dispersion waves whereas dissipative errors are hardly affected. Reducing the spatial discretization parameter ∆t minimizes dissipative errors whereas here dispersive errors are hardly affected. The best results are obtained when the ratio of both parameters in terms of the Courant number is C ≈ 0.5. Finally it is shown that the relative radius ρ is best chosen with ρ ≈ 2. Under these conditions the PCMFS technique reveals accurate results for several test cases.