In this paper, we construct and study an exactly well‐balanced positivity‐preserving nonstaggered central scheme for shallow water flows in open channels with irregular geometry and nonflat bottom topography. We introduce a novel discretization of the source term based on hydrostatic reconstruction to obtain the exactly well‐balanced property for the still water steady‐state solution even in the presence

A highly accurate method for simulating surfactant‐covered droplets in two‐dimensional Stokes flow with solid boundaries is presented. The method handles both periodic channel flows of arbitrary shape and stationary solid constrictions. A boundary integral method together with a special quadrature scheme is applied to solve the Stokes equations to high accuracy, also for closely interacting droplets

The Lattice Boltzmann Method has been widely adopted to simulate flow in porous media. The choice of appropriate boundary schemes is essential to achieve simulation accuracy, however, the criteria for the most suitable boundary treatment in the simulation of flow in porous media flow remain unresolved. Here, three types of the most commonly used boundary conditions are tested: interpolation bounce

In this paper, we develop a higher order approximation for the generalized fractional derivative that includes a scale function z(t) and a weight function w(t). This is then used to solve a generalized fractional diffusion problem numerically. The stability and convergence analysis of the numerical scheme are conducted by the energy method. It is proven that the temporal convergence order is 3 and

In this paper, adaptive mesh refinement (AMR) is performed to simulate flow around both stationary and moving boundaries. The finite‐difference approach is applied along with a sharp interface immersed boundary (IB) method. The Lagrangian polynomial is employed to facilitate the interpolation from a coarse to a fine grid level, while a weighted‐average formula is used to transfer variables inversely

This paper presents an improved fifth‐order finite difference weighted essentially non‐oscillatory (WENO) scheme to solve Hamilton‐Jacobi equations. A new type of nonlinear weights are introduced with the construction of local smoothness indicators on each local stencil that are measured with the help of generalized undivided differences in L1‐norm. A novel global smoothness measurement is also constructed

Combining the Navier–Stokes systems with Neumann (or natural) boundary condition to characterize a fluid flow is frequent. The popular projection (or pressure correction) methods inspired by Chorin and Temam are not well adapted to such boundary condition which translate in loss of accuracy. If some alternative projection methods have been proposed to reduce the accuracy loss due to the Neumann condition

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In this paper, we present an improved third‐order finite difference weighted essentially non‐oscillatory scheme to promote the order of convergence at critical points for the hyperbolic conservation laws. The improved WENO scheme is a extension of WENO‐ZQ scheme.23 However, the global smoothness indicator has a little different from WENO‐ZQ scheme. In this follow‐up paper, a convex combination of a

The dynamics of membranes, shells, and capsules in fluid flow has become an active research area in computational physics and computational biology. The small thickness of these elastic materials enables their efficient approximation as a hypersurface, which exhibits an elastic response to in‐plane stretching and out‐of‐plane bending, possibly accompanied by a surface tension force. In this work, we

In the recent decade, the meshless methods have been handled for solving most of PDEs due to easiness of the meshless methods. One of the popular meshless methods is the element free Galerkin (EFG) method that was first proposed for solving some problems in the solid mechanics. The test and trial functions of the EFG are based upon the special basis. Recently, some modifications have been developed