Local radial basis function collocation method along with explicit time stepping for hyperbolic partial differential equations

  • Authors:

  • Siraj-Ul-Islam;R. Vertnik;B. ŠArler

  • Affiliations:

  • Laboratory for Multiphase Processes, University of Nova Gorica, Vipavska 13, SI-5000 Nova Gorica, Slovenia and University of Engineering and Technology Peshawar, Pakistan;Technical Development, Štore Steel d.o.o., elezarska 3, SI-3220 Štore, Slovenia;Laboratory for Multiphase Processes, University of Nova Gorica, Vipavska 13, SI-5000 Nova Gorica, Slovenia

  • Venue:
  • Applied Numerical Mathematics
  • Year:
  • 2013

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Abstract

This paper tackles an improved Localized Radial Basis Functions Collocation Method (LRBFCM) for the numerical solution of hyperbolic partial differential equations (PDEs). The LRBFCM is based on multiquadric (MQ) Radial Basis Functions (RBFs) and belongs to a class of truly meshless methods which do not need any underlying mesh. This method can be implemented on a set of uniform or random nodes, without any a priori knowledge of node to node connectivity. We have chosen uniform nodal arrangement due their suitability and better accuracy. Five nodded domains of influence are used in the local support for the calculation of the spatial partial derivatives. This approach results in a small interpolation matrix for each data center and hence the time integration has comparatively low computational cost than the related global method. Different sizes of domain of influence i.e. m=5,13 are considered. Shape parameter sensitivity of MQ is handled through scaling technique. The time derivative is approximated by first order forward difference formula. An adaptive upwind technique is used for stabilization of the method. Capabilities of the LRBFCM are tested by applying it to one- and two-dimensional benchmark problems with discontinuities, shock pattern and periodic initial conditions. Performance of the LRBFCM is compared with analytical solution, other numerical methods and the results reported earlier in the literature. We have also made comparison with implicit first order time discretization and first order upwind spatial discretization (FVM1) and implicit second order time discretization and first order upwind spatial discretization (FVM2) as well. Accuracy of the method is assessed as a function of time and space. Numerical convergence is also shown for both one- and two-dimensional test problems. It has been observed that the proposed method is more efficient in terms of less memory requirement and less computational efforts due to one time inversion of 5x5 (size of local domain of influence) coefficient matrix. The results obtained through LBRFCM are stable and comparable with the existing methods for a variety of problems with practical applications.