Convergence analysis of sectional methods for solving aggregation population balance equations: The fixed pivot technique

Particle‐resolved simulations of turbulent particle‐laden flows provide a powerful research tool to explore detailed flow physics at all scales. However, efficient particle‐resolved simulations for wall‐bounded turbulent particle‐laden flows remain a challenging task. In this article, we develop a simulation approach for a turbulent channel flow laden with finite‐size particles on a nonuniform mesh by combining the discrete unified gas kinetic scheme (DUGKS) and the immersed boundary method (IBM). The standard discrete delta function was modified according to reproducible kernel particle method to take into account mesh non‐uniformity and correctly conserve force moments. Simulation results based on uniform and nonuniform meshes are compared to validate and examine the accuracy of the nonuniform mesh DUGKS‐IBM. Finally, the computational performance of the nonuniform mesh DUGKS‐IBM is discussed.

A high accuracy hybrid method for two-dimensional Navier–Stokes equations

Front propagation is a challenge in numerical modeling, particularly for multi-fluid or multi-material systems requiring clear material separation. The GRU (Glimm Random Update) scheme, inspired by Glimm’s method, has been developed to handle sharp fronts on unstructured 2D/3D meshes. It achieves convergence rates numerically measured between 0 . 8 and 0 . 9 , which are higher than those of classical first-order schemes, typically limited to 0 . 5 in the presence of contact discontinuities. Previous work established convergence in probability for planar fronts with uniform velocity but relied on random sequences, whereas practical implementations use low-discrepancy sequences. This study aims to establish new theoretical convergence results for the GRU scheme for 1D non-uniform meshes and uniform front velocity conditions, extending beyond uniform meshes considered previously. More precisely, we prove convergence in probability of order close to one-half when using random sequences on non-uniform meshes, and exact first-order convergence in probability when using low-discrepancy sequences (deterministic) on uniform meshes. Partial results are also obtained for non-uniform meshes, which are of significant practical interest. These findings provide insights into the scheme’s behavior with both random and low-discrepancy sequences. Numerical tests are presented to illustrate the theoretical results, with applications to non-uniform meshes, highlighting the scheme’s practical relevance.

This paper is concerned with computational mesh of the Marker and Cell method. The comutational stability of the finite difference equations which is based on the MAC method, using non-uniform mesh, is examined with respect to trucation errors and energy conservance of the finite difference equation. And restrictions for accurate finite difference calculations, using Mesh Reynolds No. and Flow-Local Reynolds No., are proposed. Those are, (1) R_m<20 (2) R_m<√<R_l> R_m : Mesh Reynolds No. R_l : Flow-Local Reynolds No. These ristrictions are examined by numerical experiments of a few flow models, and proved to be useful for both uniform and non-uniform computational meshes of the finite difference equations.

Observations on the fifth-order WENO method with non-uniform meshes

Alternating direction implicit approach for the two-dimensional time fractional nonlinear Klein–Gordon and Sine–Gordon problems

The main aim of this paper is to construct an efficient Galerkin–Legendre spectral approximation combined with a finite difference formula of L1 type to numerically solve the generalized nonlinear fractional Schrodinger equation with both space- and time-fractional derivatives. We discretize the Riesz space-fractional derivative using the Legendre–Galerkin spectral method and the time-fractional derivative using the L1 scheme on nonuniform meshes. The stability and convergence analyses of the numerical scheme are studied in detail. The scheme is unconditionally stable and convergent of $$\min \{\kappa \theta ,2-\theta \}$$ order convergence in time and of spectral accuracy in space, where $$\theta $$ is the order of fractional derivative and $$\kappa $$ is the grading mesh parameter. To verify the efficiency of the proposed algorithm, two numerical test problems are performed with convergence and error analysis.

Local derivative post-processing for the discontinuous Galerkin method

Accurate and efficient approximations for generalized population balances incorporating coagulation and fragmentation

Supraconvergent cell-centered scheme for two dimensional elliptic problems

This article presents a new methodology to compute numerical dispersion error. The analysis here presented is not restricted to uniform structured meshes nor linear discrete operators as it does not rely on sinusoids to compute the associated error. When using uniform meshes, the results obtained with the present method collapse onto the obtained with the classic one via an easy change of basis. If non-uniform meshes are used, a new kind of results are obtained which shed some light onto the role stretching has on dispersion error.

Numerical modeling of seismic waves using frequency-adaptive meshes

Numerical solution of a time-space fractional Fokker Planck equation with variable force field and diffusion

A second-order ADI difference scheme based on non-uniform meshes for the three-dimensional nonlocal evolution problem

In this work, we present a two-dimensional mixed-hybrid finite element model of variably saturated flow on unstructured triangular meshes. Velocities are approximated using lowest order Raviart-Thomas (RTo) elements with piecewise constant pressure. The resulting nonlinear systems of algebraic equations are solved using Picard or Newton iterations. Theroretical superconvergence properties of the RTo approach are experimented on analytical sample problems. It is shown that second order convergence on the pressure head and velocity fields can be obtained at particular points of the triangulation also on nonuniform meshes. Simulations on a realistic sample test show that the Newton approach achieves fast convergence if a good initial guess is provided by either the Picard scheme or relaxation techniques.