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Lattice Boltzmann method (LBM) is a powerful fluid flow solver. Using this method to solve other PDEs might be a difficult task. The first challenge is to find a suitable local equilibrium distribution function (EDF) capable of recovering the desired PDE. The next difficulty arises from the explicit nature of LBM. The conditional stability of the LBM algorithm affects the numerical solution accuracy. Damped Kuramoto–Sivashinsky (DKS) equation is a fourth–order PDE that recently challenged many numerical methods' abilities. This equation is highly sensitive to a parameter that causes three states of solutions in a small interval of freedom. In this paper, we challenged LBM to solve the two–dimensional DKS equation by finding EDF using the Chapman–Enskog analysis up to the fourth–order. Also, the von Neumann analysis and a simple genetic algorithm are applied to find reliable values for the free parameters. Furthermore, a modification on image-based ghost node method is proposed for implementation of the boundary conditions in the complex geometries. [ABSTRACT FROM AUTHOR]

The divergence-free generalized moving least squares (Div-Free GMLS) approximation has recently been utilized to solve some incompressible fluid flows problems. In our recent work (Mohammadi and Dehghan (2021) [28]), we have presented its formulation more precisely, and also the error estimates of derivatives have been carried out in L ∞ (Ω) , where Ω ⊂ R d is a bounded set satisfying an interior cone condition. However, the error estimates of this vector-valued approximation in Sobolev spaces are not done. So, in this paper we make the error estimates of Div-Free GMLS derivatives approximations in L q (Ω) , where 1 ≤ q ≤ ∞ , using a stable local divergence-free polynomial reproduction property. Note that, the method is a direct approximants of exact derivatives of a divergence-free vector field, which possesses the optimal rates of convergence. This vector-valued technique can also be developed to find the numerical solution of the incompressible fluid flows problems easier than the other available mesh-dependent methods. Finally, we have shown how the proposed approximation can recover the velocity field variable of the well-known Darcy's problem in a two-dimensional space. • The error analysis of the div-free GMLS derivatives approximations has been released on the Sobolev spaces. • The method can approximate the vector field satisfying the div-free property with the order of O (h m + 1 − | α |). • The developed vector-valued approximation satisfies analytically the div-free property for all polynomials belong to P m. • We have shown how the div-free GMLS approximation can recover the velocity field variable related to the Darcy's problem. [ABSTRACT FROM AUTHOR]

This paper introduces an extension of a fully discrete matrix–vector form (MVF) for the lattice Boltzmann method (LBM) to handle the multiple relaxation time parameter (MRT) LBM. The proposed approach offers a more efficient and practical framework for simulating fluid flows, with the added benefit of being able to handle complex geometries using the image-based ghost (IBG) method. The advantages and limitations of this approach are discussed, including its simplicity in linear algebraic extension, parallel computing capability, computational speed, and memory usage. The results of numerical experiments demonstrate the improved computational efficiency of the proposed method, highlighting its potential for future applications in various fields of computational fluid dynamics. [ABSTRACT FROM AUTHOR]

The incompressible Navier-Stokes equation can be used for modeling and simulation of fluid flow. One of the important mathematical models for simulating the pollution in the urban street canyons is the pollutant transition equation (PTE) which it is obtained from the incompressible Navier-Stokes equation. The current paper is devoted to propose a new meshless numerical procedure e.g. gradient smoothing method (GSM). First, the time derivative is approximated by the finite difference scheme. Then, the space derivative is discretized by the gradient smoothing method. Also, the proper orthogonal decomposition (POD) approach is utilized to reduce the used CPU time. Several examples with real world application are solved to verify the efficiency of the developed numerical procedure. [ABSTRACT FROM AUTHOR]

We develop a meshless numerical procedure to simulate the groundwater equation (GWE). The used technique is based on the interpolating element free Galerkin (IEFG) method. The interpolating moving least squares (IMLS) approximation produces a set of functions such that they are well-known as " shape functions ". The IEFG technique employs the shape functions of IMLS approximation. The shape functions of IMLS approximation vanish on the boundary and also they satisfy the property of the Kronecker Delta function. Thus, Dirichlet boundary conditions can be exactly imposed. In this paper, we check the unconditional stability and convergence of the proposed numerical scheme based on the energy method. The numerical results confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]