Solving helmholtz equation by least-square collocation method based on reproducing kernel particle method

Helmholtz equation often arises while solving boundaryvalue problems of partial differential equation by eigen function method. Inphysics, Helmholtz equation represents a stationary state of vibration inthe fields of mechanics, acoustics and electro-magnetics. In this paper, aleast-square collocation formulation for solving Helmholtz equation withDirichlet and Neumann boundary conditions was established. The unknowninterpolated functions were first constructed based on reproducing kernelparticle method and Helmholtz equation was then discretized by pointcollocation method. The variance errors of unknown function in each discretepoint are minimized by a least-square scheme to arrive at the finalsolution. To verify the proposed method, a wave propagation problem and aboundary layer problem of Helmholtz equation were solved. Numerical resultsby the present approach are compared with exact solutions and those bysmooth particle hydrodynamics (SPH) method. Numerical examples show that thepresent method displays better accuracy and convergence than the classicalSPH method for the same density of discrete points.