A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations.

*(English)*Zbl 0999.65135Summary: We present a thorough numerical comparison between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of boundary value problems for partial differential equations. A series of test examples was solved with these two schemes, different problems with different type of governing equations, and boundary conditions. Particular emphasis was paid to the ability of these schemes to solve the steady-state convection-diffusion equation at high values of the Péclet number.

From the examples tested in this work, it was observed that the system of algebraic equations obtained with the symmetric method was in general simpler to solve than the one obtained with the unsymmetric method and that the resulting algorithm performs better. However, the unsymmetric method has the advantage of being simpler to implement.

Two main features about the results obtained in this work are worthy of special attention: first, with the symmetric method it was possible to solve convection-diffusion problems at a very high Péclet number without the need of any artificial damping term, and second, with these two approaches, symmetric and unsymmetric, it is possible to impose free boundary conditions for problems in unbounded domains.

From the examples tested in this work, it was observed that the system of algebraic equations obtained with the symmetric method was in general simpler to solve than the one obtained with the unsymmetric method and that the resulting algorithm performs better. However, the unsymmetric method has the advantage of being simpler to implement.

Two main features about the results obtained in this work are worthy of special attention: first, with the symmetric method it was possible to solve convection-diffusion problems at a very high Péclet number without the need of any artificial damping term, and second, with these two approaches, symmetric and unsymmetric, it is possible to impose free boundary conditions for problems in unbounded domains.

##### MSC:

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

##### Keywords:

numerical examples; symmetric radial basis function; mesh free technique; numerical comparison; collocation methods; convection-diffusion equation; symmetric method; algorithm; unsymmetric method; high Péclet number; free boundary conditions; unbounded domains
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\textit{H. Power} and \textit{V. Barraco}, Comput. Math. Appl. 43, No. 3--5, 551--583 (2002; Zbl 0999.65135)

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