Nonuniform Rectangular Grid Research Articles

Stability and convergence based on the finite difference method for the nonlinear fractional cable equation on non-uniform staggered grids

In this article, a block-centered finite difference method for the nonlinear fractional cable equation is introduced and analyzed. The unconditional stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with optimal order of convergence O(Δtα+h2+k2) both for pressure and velocity are established on non-uniform rectangular grids, where α=min⁡{1+γ1,1+γ2}, Δt,h and k are the step sizes in time, space in x- and y-direction. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

A block-centered finite difference method for the nonlinear Sobolev equation on nonuniform rectangular grids

In this article, a block-centered finite difference method for the nonlinear Sobolev equation is introduced and analyzed. The stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with superconvergence O(Δt+h2+k2) for scalar unknown p, its gradient u and its flux q are established on nonuniform rectangular grids, where Δt, h and k are the step sizes in time, space in x- and y-direction. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

A block-centred finite difference method for the distributed-order differential equation with Neumann boundary condition

ABSTRACTIn this article, a block-centred finite difference method for the distributed-order differential equation with Neumann boundary condition is introduced and analysed. The stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with optimal order of convergence both for pressure and velocity are established on non-uniform rectangular grids, where and σ are the step sizes in time, space in x- and y-direction, and distributed order. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

A block-centered finite difference method for the distributed-order time-fractional diffusion-wave equation

In this article, a block-centered finite difference method for the distributed-order time-fractional diffusion-wave equation with Neumann boundary condition is introduced and analyzed. The unconditional stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with optimal order of convergence O(Δt1+σ/2+h2+k2+σ2) both for pressure and velocity are established on non-uniform rectangular grids, where Δt,h,k and σ are the step sizes in time, space in x- and y-direction, and distributed order. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

Development and analysis of Crank‐Nicolson scheme for metamaterial Maxwell's equations on nonuniform rectangular grids

In this paper, we develop a fully implicit scheme on staggered grids to solve the Maxwell's equations when Drude metamaterial is involved. Unconditional stability and optimal error estimate of the scheme are proved. Numerical results are provided to support the theoretical analysis, and used to demonstrate the applicability of the scheme to simulate the complicated backward wave propagation phenomenon occurring in metamaterials.

A block-centered finite difference method for an unsteady asymptotic coupled model in fractured media aquifer system

A block-centered finite difference method is proposed for solving an unsteady asymptotic coupled model, in which the flow is governed by Darcy’s law both in the one-dimensional fracture and two-dimensional porous media. The second-order error estimates in discrete norms are derived on nonuniform rectangular grids for both pressure and velocity. The numerical scheme can be extended to nonmatching spatial and temporal grids without loss of accuracy. Numerical experiments are performed to verify the efficiency and accuracy of the proposed method. It is shown that the pressure and velocity are discontinuous across the fracture-interface and the fracture indeed acts as the fast pathway or geological barrier in the aquifer system.

Analysis of a regularized model for the isothermal two-component mixture with the diffuse interface

AbstractA regularized system of equations describing a flow of isothermal two-component mixture with diffuse interface is studied. The equation of energy balance and its corollary, i.e., the law of non-increasing of the total energy are derived under general assumptions on the Helmholtz free energy of the mixture. Necessary and sufficient conditions for linearized stability of constant solutions are obtained (in particular case). A difference approximation of the problem is constructed in the two-dimensional periodic case on a nonuniform rectangular grid. The results of numerical experiments demonstrate a qualitative well-posedness of the problem and the applicability of the criterion of linearized stabilization in the original nonlinear formulation.

A high-order fully conservative block-centered finite difference method for the time-fractional advection–dispersion equation

Based on the weighted and shifted Grünwald–Letnikov difference operator, a new high-order block-centered finite difference method is derived for the time-fractional advection–dispersion equation by introducing an auxiliary flux variable to guarantee full mass conservation. The stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with optimal order of convergence O(Δt3+h2+k2) both for solute concentration and the auxiliary flux variable are established on non-uniform rectangular grids, where Δt, h and k are the step sizes in time, space in x- and y-direction. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

A block‐centered finite difference method for fractional Cattaneo equation

In this article, a block‐centered finite difference method for fractional Cattaneo equation is introduced and analyzed. The unconditional stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norm with optimal order of convergence both for pressure and velocity are established on nonuniform rectangular grids. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations

The multidimensional quasi-gasdynamic system written in the form of mass, momentum, and total energy balance equations for a perfect polytropic gas with allowance for a body force and a heat source is considered. A new conservative symmetric spatial discretization of these equations on a nonuniform rectangular grid is constructed (with the basic unknown functions—density, velocity, and temperature—defined on a common grid and with fluxes and viscous stresses defined on staggered grids). Primary attention is given to the analysis of entropy behavior: the discretization is specially constructed so that the total entropy does not decrease. This is achieved via a substantial revision of the standard discretization and applying numerous original features. A simplification of the constructed discretization serves as a conservative discretization with nondecreasing total entropy for the simpler quasi-hydrodynamic system of equations. In the absence of regularizing terms, the results also hold for the Navier–Stokes equations of a viscous compressible heat-conducting gas.

A Low-Dispersion Realization of a Rectangular Grid With PITD Method Through Artificial Anisotropy

Through using a nonuniform rectangular grid in the time-domain algorithm, the computational efficiency can be improved obviously, but the numerical anisotropic dispersion error is seriously deteriorative. In this letter, a rectangular grid with the precise-integration time-domain method through artificial anisotropy is proposed to reduce the numerical dispersion error of a rectangular grid. Both the stability condition and the numerical dispersion equation are obtained analytically, and the numerical anisotropic dispersion of the proposed method is examined in detail. It is found that the numerical dispersion error can be reduced obviously and can also be made nearly independent of the time-step size. The numerical experiments validate and verify that the proposed method is of higher accuracy and efficiency.

Block-centered finite difference methods for general Darcy–Forchheimer problems

Block-centered finite difference methods are constructed to solve the general Darcy–Forchheimer problems with Neumann boundary conditions, in which the velocity and pressure can be approximated simultaneously. We demonstrate that with sufficiently smooth analytical solution, the errors for both pressure and velocity in discrete L2-norms are second-order accurate on a nonuniform rectangular grid. Numerical experiments carried out using the scheme show the consistency of the convergence rates of our method with the theoretical analysis.

A Block-Centered Finite Difference Method for Slightly Compressible Darcy–Forchheimer Flow in Porous Media

A block-centered finite difference method is introduced to solve an initial and boundary value problem for a nonlinear parabolic equation to model the slightly compressible flow in porous media, in which the velocity–pressure relation is described by Darcy–Forchheimer’s Law. The method can be thought as the lowest order Raviart–Thomas mixed element method with proper quadrature formulation. By using the method the velocity and pressure can be approximated simultaneously. We established the second-order error estimates for pressure and velocity in proper discrete norms on non-uniform rectangular grid. No time-step restriction is needed for the error estimates. The numerical experiments using the scheme show that the convergence rates of the method are in agreement with the theoretical analysis.

A Two-Grid Block-Centered Finite Difference Method for the Nonlinear Time-Fractional Parabolic Equation

In this article, a two-grid block-centered finite difference scheme is introduced and analyzed to solve the nonlinear time-fractional parabolic equation. This method is considered where the nonlinear problem is solved only on a coarse grid of size H and a linear problem is solved on a fine grid of size h. Stability results are proven rigorously. Error estimates are established on non-uniform rectangular grid which show that the discrete $$L^{\infty }(L^2)$$ and $$L^2(H^1)$$ errors are $$O(\triangle t^{2-\alpha }+h^2+H^3)$$ . Finally, some numerical experiments are presented to show the efficiency of the two-grid method and verify that the convergence rates are in agreement with the theoretical analysis.

Numerical analysis of the nonlinear plane Couette-flow problem of a rarefied gas for hard-sphere molecules

The plane Couette flow of a rarefied gas is investigated on the basis of the Boltzmann equation over the wide range of Knudsen and Mach numbers. The velocity distribution function and its first thirteen moments are obtained from the accurate numerical solution based on the projection discrete-velocity method extended for nonuniform rectangular velocity grids. The DSMC simulation is used to reinforce the obtained results. The nonlinear Hilbert-type asymptotic solution for a slightly rarefied gas is constructed and also included in the comparison. For this purpose, some additional transport coefficients for hard-sphere molecules are evaluated.

On new spatial discretization of the multidimensional quasi-gasdynamic system of equations with nondecreasing total entropy

The multidimensional quasi-gasdynamic system of equations written in the form of mass, momentum, and total energy balance equations for a perfect polytropic gas with allowance for a body force and a heat source is considered. A new conservative symmetric spatial discretization on a nonuniform rectangular grid is constructed for this system. The basic unknown functions (density, velocity, and temperature) are defined on a common grid, while the fluxes and viscous stresses, on staggered grids. The discretization is specially constructed so that the total entropy does not decrease, which is achieved by applying numerous original features.

A block-centered finite difference method for general non-Darcy flow in porous media

A block-centered finite difference scheme is introduced to solve the nonlinear equation of non-Darcy model, in which it can maintain local mass conservation and the velocity and pressure can be approximated simultaneously. The second-order error estimates in discrete $l^2$ norm for both pressure and velocity are established on a nonuniform rectangular grid. Numerical experiments using the scheme show the consistency of the convergence rates of our method with the theoretical analysis.

A parallel CGS block-centered finite difference method for a nonlinear time-fractional parabolic equation

In this article, a parallel Conjugate Gradient Squared (CGS) block-centered finite difference scheme is introduced and analyzed to cast about the numerical solution of a nonlinear time-fractional parabolic equation with the Neumann condition and a nonlinear reaction term. The unconditionally stable result, which just depends on initial value and source item, is derived. Some a priori estimates of discrete L2-norm with optimal order of convergence O(Δt2−α+h2+k2) with pressure and velocity are established on both uniform and non-uniform rectangular grids, where Δt is the time step, h and k are maximal mesh sizes of x and y-directional grids. In our model, because the simulation is iterated over a series of time steps, it is most beneficial if all the calculation units in a time step can be run separately. In CGS algorithm, adding OpenMP instructions to the circulation calculations in an iterative step can realize parallel computing. To examine the efficiency and accuracy of the proposed method, numerical experiments using the schemes are studied. The results clearly show the benefit of using the proposed approach in terms of execution time reduction and speedup with respect to the sequential running in a single thread.

A two-grid block-centered finite difference method for nonlinear non-Fickian flow model

In this paper, a two-grid block-centered finite difference scheme is introduced and analyzed to solve the nonlinear parabolic integro-differential equation arising in modeling non-Fickian flow in porous media. This method is considered where the nonlinear problem is solved only on a coarse grid of size H and a linear problem is solved on a fine grid of size h. Error estimates are established on non-uniform rectangular grid which show that the discrete L∞(L2) and L2(H1) errors are O(▵t+h2+H3). Finally, some numerical experiments are presented to show the efficiency of the two-grid method and verify that the convergence rates are in agreement with the theoretical analysis.

On conservative spatial discretizations of the barotropic quasi-gasdynamic system of equations with a potential body force

A multidimensional barotropic quasi-gasdynamic system of equations in the form of mass and momentum conservation laws with a general gas equation of state p = p(ρ) with p′(ρ) > 0 and a potential body force is considered. For this system, two new symmetric spatial discretizations on nonuniform rectangular grids are constructed (in which the density and velocity are defined on the basic grid, while the components of the regularized mass flux and the viscous stress tensor are defined on staggered grids). These discretizations involve nonstandard approximations for ∇p(ρ), div(ρu), and ρ. As a result, a discrete total mass conservation law and a discrete energy inequality guaranteeing that the total energy does not grow with time can be derived. Importantly, these discretizations have the additional property of being well-balanced for equilibrium solutions. Another conservative discretization is discussed in which all mass flux components and viscous stresses are defined on the same grid. For the simpler barotropic quasi-hydrodynamic system of equations, the corresponding simplifications of the constructed discretizations have similar properties.