Radial point interpolation collocation method (RPICM) for partial differential equations. (English) Zbl 1083.65108
Summary: The authors present a truly meshfree method referred to as radial point interpolation collocation method (RPICM) for solving partial differential equations. This method is different from the existing point interpolation method that is based on the Galerkin weak-form. Because it is based on the collocation scheme no background cells are required for numerical integration. Radial basis functions are used in the work to create shape functions.
A series of test examples are numerically analysed using the present method, including 1D and 2D partial differential equations, in order to test the accuracy and efficiency of the proposed schemes. Several aspects are numerically investigated, including the choice of shape parameter \(c\) with can greatly affect the accuracy of the approximation; the enforcement of additional polynomial terms; and the application of the Hermite-type interpolation which makes use of the normal gradient on the Neumann boundary for the solution of partial differential equations with Neumann boundary conditions.
Particular emphasis is on an efficient scheme, namely Hermite-type interpolation for dealing with Neumann boundary conditions. The numerical results demonstrate that good improvement on accuracy can be obtained after using Hermite-type interpolation. The \(h\)-convergence rates are also studied for RPICM with different forms of basis functions and different additional terms.
A series of test examples are numerically analysed using the present method, including 1D and 2D partial differential equations, in order to test the accuracy and efficiency of the proposed schemes. Several aspects are numerically investigated, including the choice of shape parameter \(c\) with can greatly affect the accuracy of the approximation; the enforcement of additional polynomial terms; and the application of the Hermite-type interpolation which makes use of the normal gradient on the Neumann boundary for the solution of partial differential equations with Neumann boundary conditions.
Particular emphasis is on an efficient scheme, namely Hermite-type interpolation for dealing with Neumann boundary conditions. The numerical results demonstrate that good improvement on accuracy can be obtained after using Hermite-type interpolation. The \(h\)-convergence rates are also studied for RPICM with different forms of basis functions and different additional terms.
MSC:
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
Keywords:
Hermite-type interpolation; numerical examples; meshfree method; radial basis functions; Neumann boundary conditions; convergenceSoftware:
Mfree2DReferences:
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