The use of proper orthogonal decomposition (POD) meshless RBF-FD technique to simulate the shallow water equations

The main objective in this paper is to propose an efficient numerical formulation for solving the time-fractional distributed-order advection–diffusion equation. First, the distributed-order term has been approximated by the Gauss quadrature rule. In the next, a finite difference approach is applied to approximate the temporal variable with convergence order $$\mathcal{O}(\tau ^{2-\alpha })$$
 as $$0<\alpha <1$$
 . Finally, to discrete the spacial dimension, an upwind local radial basis function-finite difference idea has been employed. In the numerical investigation, the effect of the advection coefficient has been studied in terms of accuracy and stability of the proposed difference scheme. At the end, two examples are studied to approve the impact and ability of the numerical procedure.

Meshless upwind local radial basis function-finite difference technique to simulate the time- fractional distributed-order advection–diffusion equation

In this investigation, we concentrate on solving the regularized long-wave (RLW) and extended Fisher–Kolmogorov (EFK) equations in one-, two-, and three-dimensional cases by a local meshless method called radial basis function (RBF)–finite-difference (FD) method. This method at each stencil approximates differential operators such as finite-difference method. In each stencil, it is necessary to solve a small-sized linear system with conditionally positive definite coefficient matrix. This method is relatively efficient and has low computational cost. How to choose the shape parameter is a fundamental subject in this method, since it has a palpable effect on coefficient matrix. We will employ the optimal shape parameter which results from algorithm of Sarra (Appl Math Comput 218:9853–9865, 2012). Then, we compare the approximate solutions acquired by RBF–FD method with theoretical solution and also with results obtained from other methods. The numerical results show that the RBF–FD method is suitable and robust for solving the RLW and EFK equations.

Local radial basis function–finite-difference method to simulate some models in the nonlinear wave phenomena: regularized long-wave and extended Fisher–Kolmogorov equations

In this paper, we use a linear combination of the shape functions of reproducing kernel particle method (RKPM) and RBFs for achieving the unknown weights into each stencil. We obtain an error bound for the new shape function. Also, in this paper, we investigate a numerical procedure based on the presented technique for solving the Vlasov---Poisson and Vlasov---Poisson---Fokker---Planck systems. The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma. The Vlasov---Poisson equations are used to describe various phenomena in plasma, in particular Landau damping and the distributions in a double layer plasma. We use the RKPM/RBF-FD technique for discretization of space direction and employ the method of lines to achieve a high-order accuracy in temporal direction. Numerical examples are reported which demonstrate the theoretical results and the efficiency of proposed scheme.

A local meshless method for solving multi-dimensional Vlasov---Poisson and Vlasov---Poisson---Fokker---Planck systems arising in plasma physics

In the current investigation, we propose two efficient meshless numerical techniques for solving two models in optic engineering, i.e. the generalized Gross–Pitaevskii equation and the generalized Zakharov system. Two local meshless methods have been employed for solving these models: local radial basis functions collocation method and direct meshless local Petrov–Galerkin method. In this paper, we discrete the space direction using the local radial basis functions collocation and direct meshless local Petrov–Galerkin techniques and to obtain high-order numerical results, we use the fourth-order exponential time differencing Runge–Kutta method for discretizing the temporal direction. The obtained numerical results are compared with some well-known numerical techniques. Moreover, several examples are given that show the acceptable accuracy and efficiency of the proposed scheme.

Numerical investigation based on direct meshless local Petrov Galerkin (direct MLPG) method for solving generalized Zakharov system in one and two dimensions and generalized Gross–Pitaevskii equation

Shape adaptive RBF-FD implicit scheme for incompressible viscous Navier–Strokes equations

In this paper, a coupled meshfree-mesh-based fluid solver is employed for flow-induced vibration problems. The fluid domain comprises of a hybrid grid which is formed by generating a body conformal meshfree nodal cloud around the solid object and a static Cartesian grid which surrounds the meshfree cloud in the far field. The meshfree nodal cloud provides flexibility in dealing with solid motion by moving and morphing along with the solid boundary without necessitating re-meshing. The Cartesian grid, on the other hand, provides improved performance by allowing the use of a computationally efficient mesh-based method. The flow equations, in arbitrary Lagrangian–Eulerian formulation, are solved by a local radial basis function in finite difference mode on moving meshfree nodes. Conventional finite differencing is used over the static Cartesian grid for flow equations in Eulerian formulation. The equations for solid motion are solved using a classic Runge–Kutta method. Closed coupling is introduced between fluid and structural solvers by using a sub-iterative prediction–correction algorithm. In order to reduce computational overhead due to sub-iterations, only near-field flow (in the meshfree zone) is solved during the inner iterations. The full fluid domain is solved during outer (time step) iterations only when the convergence at the solid–fluid interface has already been reached. In the meshfree zone, adaptive sizing of the influence domain is introduced to maintain suitable number of neighbouring particles. The use of a hybrid grid has been found to be useful in improving the computational performance by faster computing over the Cartesian grid as well as by reducing the number of computations in the fluid domain during fluid–solid coupling. The solution scheme was tested for problems relating to flow-induced cylindrical and airfoil vibration with one and two degrees of freedom. The results are found to be in good agreement with previous work and experimental results.

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K. Djijdeli

Papers

Shape adaptive RBF-FD implicit scheme for incompressible viscous Navier–Strokes equations

A coupled meshfree-mesh-based solution scheme on hybrid grid for flow-induced vibrations