Error estimates of a linear decoupled Euler–FEM scheme for a mass diffusion model

We present error estimates of a linear fully discrete scheme for a three-dimensional mass diffusion model for incompressible fluids (also called Kazhikhov–Smagulov model). All unknowns of the model (velocity, pressure and density) are approximated in space by C ⁰-finite elements and in time an Euler type scheme is used decoupling the density from the velocity–pressure pair. If we assume that the velocity and pressure finite-element spaces satisfy the inf–sup condition and the density finite-element space contains the products of any two discrete velocities, we first obtain point-wise stability estimates for the density, under the constraint lim_(h,k)→0 h/k = 0 (h and k being the space and time discrete parameters, respectively), and error estimates for the velocity and density in energy type norms, at the same time. Afterwards, error estimates for the density in stronger norms are deduced. All these error estimates will be optimal (of order O(h+k){\mathcal{O}(h+k)}) for regular enough solutions without imposing nonlocal compatibility conditions at the initial time. Finally, we also study two convergent iterative methods for the two problems to solve at each time step, which hold constant matrices (independent of iterations).     

One‐level and two‐level finite element methods for the Boussinesq problem

Previous works on the convergence of numerical methods for the Boussinesq problem were conducted, while the optimal L²‐norm error estimates for the velocity and temperature are still lacked. In this paper, the backward Euler scheme is used to discrete the time terms, standard Galerkin finite element method is adopted to approximate the variables. The MINI element is used to approximate the velocity and pressure, the temperature field is simulated by the linear polynomial. Under some restriction on the time step, we firstly present the optimal L² error estimates of approximate solutions. Secondly, two‐level method based on Stokes iteration for the Boussinesq problem is developed and the corresponding convergence results are presented. By this method, the original problem is decoupled into two small linear subproblems. Compared with the standard Galerkin method, the two‐level method not only keeps good accuracy but also saves a lot of computational cost. Finally, some numerical examples are provided to support the established theoretical analysis.     

Convergence of the Grünwald–Letnikov scheme for time-fractional diffusion

Using bivariate generating functions, we prove convergence of the Grünwald–Letnikov difference scheme for the fractional diffusion equation (in one space dimension) with and without central linear drift in the Fourier–Laplace domain as the space and time steps tend to zero in a well-scaled way. This implies convergence in distribution (weak convergence) of the discrete solution towards the probability of sojourn of a diffusing particle. The difference schemes allow also interpretation as discrete random walks. For fractional diffusion with central linear drift we show that in the Fourier–Laplace domain the limiting ordinary differential equation coincides with that for the solution of the corresponding diffusion equation.     

Local discontinuous Galerkin methods with explicit Runge‐Kutta time marching for nonlinear carburizing model

A fully discrete local discontinuous Galerkin (LDG) method coupled with 3 total variation diminishing Runge‐Kutta time‐marching schemes, for solving a nonlinear carburizing model, will be analyzed and implemented in this paper. On the basis of a suitable numerical flux setting in the LDG method, we obtain the optimal error estimate for the Runge‐Kutta–LDG schemes by energy analysis, under the condition τ ≤ λh², where h and τ are mesh size and time step, respectively, λ is a positive constant independent of h. Numerical experiments are presented to verify the accuracy and capability of the proposed schemes. For the carburizing diffusion processes of steel and the diffusion simulation for Cu‐Ni system, the numerical results show good agreement with the experimental results.     

Stabilized DDFV schemes for stokes problem with variable viscosity on general 2D meshes

“Discrete Duality Finite Volume” schemes (DDFV for short) on general meshes are studied here for Stokes problems with variable viscosity with Dirichlet boundary conditions. The aim of this work is to analyze the well‐posedness of the scheme and its convergence properties. The DDFV method requires a staggered scheme, the discrete unknowns, the components of the velocity and the pressure, are located on different nodes. The scheme is stabilized using a finite volume analogue to Brezzi‐Pitkäranta techniques. This scheme is proved to be well‐posed on general meshes and to be first order convergent in a discrete H¹ ‐norm and a discrete L² ‐norm for respectively the velocity and the pressure. Finally, numerical experiments confirm the theoretical prediction, in particular on locally refined non conformal meshes. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1666–1706, 2011     

The Crank‐Nicolson/Adams‐Bashforth scheme for the time‐dependent Navier‐Stokes equations with nonsmooth initial data

In this article, we study the stability and convergence of the Crank‐Nicolson/Adams‐Bashforth scheme for the two‐dimensional nonstationary Navier‐Stokes equations with a nonsmooth initial data. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the implicit Crank‐Nicolson scheme for the linear terms and the explicit Adams‐Bashforth scheme for the nonlinear term. Moreover, we prove that the scheme is almost unconditionally stable for a nonsmooth initial data u₀ with div u₀ = 0, i.e., the time step τ satisfies: τ ≤ C₀ if u₀ ∈ H¹ ∩ L^∞; τ |log h| ≤ C₀ if u₀ ∈ H¹ for the mesh size h and some positive constant C₀. Finally, we obtain some error estimates for the discrete velocity and pressure under the above stability condition. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 155‐187, 2012     

Convergence analysis and computational testing of the finite element discretization of the Navier–Stokes alpha model

This report performs a complete analysis of convergence and rates of convergence of finite element approximations of the Navier–Stokes‐α (NS‐α) regularization of the NSE, under a zero‐divergence constraint on the velocity, to the true solution of the NSE. Convergence of the discrete NS‐α approximate velocity to the true Navier–Stokes velocity is proved and rates of convergence derived, under no‐slip boundary conditions. Generalization of the results herein to periodic boundary conditions is evident. Two‐dimensional experiments are performed, verifying convergence and predicted rates of convergence. It is shown that the NS‐α‐FE solutions converge at the theoretical limit of O(h²) when choosing α = h, in the H¹ norm. Furthermore, in the case of flow over a step the NS‐α model is shown to resolve vortex separation in the recirculation zone. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Convergence analysis of an implicit fractional-step method for the incompressible Navier–Stokes equations

In this paper, an implicit fractional-step method for numerical solutions of the incompressible Navier–Stokes equations is studied. The time advancement is decomposed into a sequence of two steps, and the first step can be seen as a linear elliptic problem; on the other hand, the second step has the structure of the Stokes problem. The two problems satisfy the full homogeneous Dirichlet boundary conditions on the velocity. At the same time, we introduce a diffusion term −θΔu in all steps of the schemes. It allows to calculate by the large time step and enhance numerical stability by choosing the proper parameter values of θ. The convergence analysis and error estimates for the intermediate velocities, the end-of step velocities and the pressure solution are derived. Finally, numerical experiments show that the feasibility and effectiveness of this method.     

Construction and Analysis of Lattice Boltzmann Methods Applied to a 1D Convection-Diffusion Equation

To solve the 1D (linear) convection-diffusion equation, we construct and we analyze two LBM schemes built on the D1Q2 lattice. We obtain these LBM schemes by showing that the 1D convection-diffusion equation is the fluid limit of a discrete velocity kinetic system. Then, we show in the periodic case that these LBM schemes are equivalent to a finite difference type scheme named LFCCDF scheme. This allows us, firstly, to prove the convergence in L ^∞ of these schemes, and to obtain discrete maximum principles for any time step in the case of the 1D diffusion equation with different boundary conditions. Secondly, this allows us to obtain most of these results for the Du Fort-Frankel scheme for a particular choice of the first iterate. We also underline that these LBM schemes can be applied to the (linear) advection equation and we obtain a stability result in L ^∞ under a classical CFL condition. Moreover, by proposing a probabilistic interpretation of these LBM schemes, we also obtain Monte-Carlo algorithms which approach the 1D (linear) diffusion equation. At last, we present numerical applications justifying these results.     

Unconditional stability and convergence of fully discrete schemes for viscous fluids models with mass diffusion

In this work we develop fully discrete (in time and space) numerical schemes for two-dimensional incompressible fluids with mass diffusion, also so-called Kazhikhov-Smagulov models. We propose at most -conformed finite elements (only globally continuous functions) to approximate all unknowns (velocity, pressure and density), although the limit density (solution of continuous problem) will have regularity. A backward Euler in time scheme is considered decoupling the computation of the density from the velocity and pressure.

Unconditional stability of the schemes and convergence towards the (unique) global in time weak solution of the models is proved. Since a discrete maximum principle cannot be ensured, we must use a different interpolation inequality to obtain the strong estimates for the discrete density, from the used one in the continuous case. This inequality is a discrete version of the Gagliardo-Nirenberg interpolation inequality in domains. Moreover, the discrete density is truncated in some adequate terms of the velocity-pressure problem.

     

The discrete duality finite volume method for a class of quasi‐Newtonian Stokes flows

In this paper, we propose a discrete duality finite volume (DDFV) scheme for the incompressible quasi‐Newtonian Stokes equation. The DDFV method is based on the use of discrete differential operators which satisfy some duality properties analogous to their continuous counterparts in a discrete sense. The DDFV method has a great ability to handle general geometries and meshes. In addition, every component of the velocity gradient can be reconstructed directly, which makes it suitable to deal with the nonlinear terms in the quasi‐Newtonian Stokes equation. We prove that the proposed DDFV scheme is uniquely solvable and of first‐order convergence in the discrete L²‐norms for the velocity, the strain rate tensor, and the pressure, respectively. Ample numerical tests are provided to highlight the performance of the proposed DDFV scheme and to validate the theoretical error analysis, in particular on locally refined nonconforming and polygonal meshes.     

Explicit iteration schemes for minimization problems arising from image denoising

The purpose of this paper is to investigate explicit iteration schemes for minimization problems arising from image denoising. In particular, we propose explicit iteration schemes based on matrix splitting. When the matrix splitting is done by the symmetric Gauss–Seidel method, we establish convergence of the scheme with no restriction on the step size of the iteration. If the matrix splitting is done by the Gauss–Seidel method, we show that the iteration scheme still converges, provided the step size of each iteration is sufficiently small.     

New energy identities and super convergence analysis of the energy conserved splitting FDTD methods for 3D Maxwell's equations

The energy‐conserved splitting finite‐difference time‐domain (EC‐S‐FDTD) method has recently been proposed to solve the Maxwell equations with second order accuracy while numerically keep the L² energy conservation laws of the equations. In this paper, the EC‐S‐FDTD scheme for the 3D Maxwell equations is proved to be energy‐conserved and unconditionally stable in the discrete H¹ norm. The EC‐S‐FDTD scheme is of second‐order accuracy both in time step and spatial steps, which suggests the super‐convergence of this scheme in the discrete H¹ norm. And the divergence of the electric field of the EC‐S‐FDTD scheme in the discrete L² norm is second‐order accurate. Numerical experiments confirm our theoretical analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

A fully discrete spectral method for fractional Cattaneo equation based on Caputo–Fabrizo derivative

Recently Caputo and Fabrizio introduced a new derivative with fractional order without singular kernel. The derivative can be used to describe the material heterogeneities and the fluctuations of different scales. In this article, we derived a new discretization of Caputo–Fabrizio derivative of order α (1 < α < 2) and applied it into the Cattaneo equation. A fully discrete scheme based on finite difference method in time and Legendre spectral approximation in space is proposed. The stability and convergence of the fully discrete scheme are rigorously established. The convergence rate of the fully discrete scheme in H¹ norm is O(τ² + N^1?m), where τ, N and m are the time‐step size, polynomial degree and regularity in the space variable of the exact solution, respectively. Furthermore, the accuracy and applicability of the scheme are confirmed by numerical examples to support the theoretical results.     

Short Note: A Note on Nonstandard Finite Difference Schemes for Reaction–Diffusion Equations Having Linear Advection

Nonstandard modified upwind difference scheme for one dimensional nonlinear reaction–diffusion equation with linear advection is given in this note. The use of a positivity condition allows the determination of a functional relation between the time and space step sizes, and it is weaker than that of the corresponding simple upwind difference scheme. Error estimate in the discrete l ^∞ norm is provided under suitable assumptions.     

A fully discrete H 1-Galerkin method with quadrature for nonlinear advection–diffusion–reaction equations

We propose and analyze a fully discrete H ¹-Galerkin method with quadrature for nonlinear parabolic advection–diffusion–reaction equations that requires only linear algebraic solvers. Our scheme applied to the special case heat equation is a fully discrete quadrature version of the least-squares method. We prove second order convergence in time and optimal H ¹ convergence in space for the computer implementable method. The results of numerical computations demonstrate optimal order convergence of scheme in H ^k for k = 0, 1, 2. Support of the Australian Research Council is gratefully acknowledged.     

Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion

We present a new a priori estimate for discrete coagulation–fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a priori estimate provides a global L²$L^2$ bound on the mass density and was previously used, for instance, in the context of reaction–diffusion equations.     

Thermodynamics / turbulence analogy modelling dissipating molecular gas flows for high Knudsen numbers

Modelling micro channel flows momentum and heat diffusion / convection are recent parameters modelling the molecule velocity distribution. Macroscopic models describe velocity and energy / enthalpie with integrals of mass increments. Using microscopic models motion and forces of a molecular flow have to be computed by models of physical properties, whose are described by statistical power moments of the molecule velocity. Therefore dilute flows have to be investigated in small channels with a mean free path length of molecules higher than the channel width of the the micro channel itself (λ₀≥H₀). Modelling this process by a continuous flow the boundary conditions have to be modified (e.g. [6]). The present model uses the statistical approximation of the molecule velocity distribution to simulate the behaviour of this discrete flow with a weighted averaged molecule velocity ∼ξ_i, its standard deviation σ and the characterisic molecule collision rate z. The number density N per volume V near one position is used for the weighting factor averaging method describing the mean molecule velocity. The present model is validated computing Poiseuille and Couette flows with different Knudsen numbers. Showing the advantages of the present model the simulation results are compared with simulation results of the wall–distance depending diffusivity model of Lockerby and Reese [4] and BGK results of a Lattice–Boltzmann simulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal point-wise error estimate of a compact difference scheme for the Klein–Gordon–Schrödinger equation

In this paper, we propose a compact finite difference scheme for computing the Klein–Gordon–Schrödinger equation (KGSE) with homogeneous Dirichlet boundary conditions. The proposed scheme not only conserves the total mass and energy in the discrete level but also is linearized in practical computation. Except for the standard energy method, a new technique is introduced to obtain the optimal convergent rate, without any restriction on the grid ratios, at the order of O(h⁴+τ²)$O(h^4+{\tau }^2)$ in the l^∞$l^{\infty }$-norm with time step τ and mesh size h. Finally, numerical results are reported to test the theoretical results.     

Asymptotic analysis and estimates of blow‐up time for the radial symmetric semilinear heat equation in the open‐spectrum case

We estimate the blow‐up time for the reaction diffusion equation u_t=Δu+ λf(u), for the radial symmetric case, where f is a positive, increasing and convex function growing fast enough at infinity. Here λ>λ^*, where λ^* is the ‘extremal’ (critical) value for λ, such that there exists an ‘extremal’ weak but not a classical steady‐state solution at λ=λ^* with ∥w(?, λ)∥_∞→∞ as 0<λ→λ^*?. Estimates of the blow‐up time are obtained by using comparison methods. Also an asymptotic analysis is applied when f(s)=e^s, for λ?λ^*?1, regarding the form of the solution during blow‐up and an asymptotic estimate of blow‐up time is obtained. Finally, some numerical results are also presented. Copyright © 2007 John Wiley & Sons, Ltd.