A stabilized meshfree reproducing kernel‐based method for convection–diffusion problems

Abstract

SUMMARYIn this paper, we develop a meshfree particle-based method for convection-diffusion problems. Discretization is performed by using piecewise constant kernels. The stabilized scheme is based on a new upwind kernel. We show that accurate and stable scheme can be obtained by using purpose-built kernels. It also shown that under some conditions the classical optimal finite difference scheme can be derived by the new method. Several numerical tests validate the method.

Numerical modeling of magneto‐hydrodynamics flows using reproducing kernel particle method

“…Problem 1: For the first problem, suppose that . In this case, the approximation of the first derivative reduces to a finite difference scheme . In this case, approximation of the first derivative for center and upwind kernels will become center difference and forward or backward difference, respectively.…”

“…For this reason, we consider the following modification of the system : where the matrix is obtained from the identity matrix I n by replacing the diagonal elements ( I n ) j j and ( I n ) k k with 0. This can be interpreted as a minimization of all the weights ω i expects ω j and ω k .…”

“…When the center kernel for the first derivative of function f is used, the solution of MHD equations is unstable for high Hartmann numbers, as can be seen in Figure . For this reason, in , an upwind kernel is introduced to approximate first derivatives in convection–diffusion equations. We use this kernel to obtain a stable scheme for MHD equations, and next, a combination of center and upwind kernels is used to obtain a stable and accurate solution for MHD equations.…”

“…In , the nearly optimal kernel was introduced for convection–diffusion problems. Because the MHD equations with the form contain two decoupled convection–diffusion problems, we use this kernel to obtain a scheme that is accurate and stable.…”

Reproducing Kernel Particle Method for Radiative Heat Transfer in 1D Participating Media

“…So the RKPM can work well over the whole domain. It has wide application in many engineering fields, such as, rolling plane strain problem [13], bucking analysis of thin plates [14], large deformation nonlinear elastic problems [15][16][17][18], metal forming problem [19][20][21][22], elastic-plastic problems [23,24], convection-diffusion problem [25][26][27], heat conduction problems [28][29][30], and fragment-impact problem [31,32]. But up to now, to the best of our knowledge, there is still lack of literature on the application of RKPM for radiative heat transfer.…”