Backward Euler Difference Scheme Research Articles

Stabilized mixed finite element method for a quasistatic Maxwell viscoelastic model

This paper considers a stabilized mixed finite element method (MFE) for a quasistatic Maxwell viscoelastic model based on the L2(Ω)×H1(Ω) variational framework. The spatial discretization uses arbitrary degree piecewise polynomials Pl−Pk(l≥0,k≥1) to approximate stress and velocity. The temporal discretization in the fully discrete method adopts a backward Euler difference scheme. We give a unified analysis to show the stability of the semi-discrete and fully discrete solutions and derive the corresponding priori error estimates. Numerical examples are provided to support the theoretical analysis.

A new class of fractional differential hemivariational inequalities with application to an incompressible Navier-Stokes system coupled with a fractional diffusion equation

This paper is devoted to the study of a new and complicated dynamical system, called a fractional differential hemivariational inequality, which consists of a quasilinear evolution equation involving the fractional Caputo derivative operator and a coupled generalized parabolic hemivariational inequality. Under certain general assumptions, existence and regularity of a mild solution to the dynamical system are established by employing a surjectivity result for weakly-weakly upper semicontinuous multivalued mappings, and a feedback iterative technique together with a temporally semi-discrete approach through the backward Euler difference scheme with quasi-uniform time-steps. To illustrate the applicability of the abstract results, we consider a nonstationary and incompressible Navier-Stokes system supplemented by a fractional reaction-diffusion equation, which is studied as a fractional hemivariational inequality. Bibliography: 57 titles.

A new class of fractional differential hemivariational inequalities with application to an incompressible Navier-Stokes system coupled with a fractional diffusion equation

This paper is devoted to the study of a new and complicated dynamical system, called a fractional differential hemivariational inequality, which consists of a quasilinear evolution equation involving the fractional Caputo derivative operator and a coupled generalized parabolic hemivariational inequality. Under certain general assumptions, existence and regularity of a mild solution to the dynamical system are established by employing a surjectivity result for weakly-weakly upper semicontinuous multivalued mappings, and a feedback iterative technique together with a temporally semi-discrete approach through the backward Euler difference scheme with quasi-uniform time-steps. To illustrate the applicability of the abstract results, we consider a nonstationary and incompressible Navier-Stokes system supplemented by a fractional reaction-diffusion equation, which is studied as a fractional hemivariational inequality.

Rothe's method for solving multi‐term Caputo–Katugampola fractional delay integral diffusion equations

This paper discusses a class of multi‐term Caputo–Katugampola fractional delay integral diffusion equations (MCKFIDEs, for short) in Hilbert spaces. A iterative scheme in interval corresponding to the MCKFIDE is introduced by using a temporally semi‐discrete method based on the backward Euler difference scheme, i.e., Rothe's method. First, we apply the iterative scheme and the ‐accretivity of operator to establish existence, uniqueness, and a priori estimate for strong solutions to an approximate problem. Based on this result, we obtain existence, regularity of the strong solution for MCKFIDEs on interval or the maximum interval . Then, we also prove that the strong solution is unique if and only if the delay boundary condition is unique on . Finally, two examples are given to illustrate the main results.

EXISTENCE OF SOLUTIONS FOR FRACTIONAL EVOLUTION INCLUSION WITH APPLICATION TO MECHANICAL CONTACT PROBLEMS

The goal of this paper is to study an evolution inclusion problem with fractional derivative in the sense of Caputo, and Clarke’s subgradient. Using the temporally semi-discrete method based on the backward Euler difference scheme, we introduce a discrete approximation system of elliptic type corresponding to the fractional evolution inclusion problem. Then, we employ the surjectivity of multivalued pseudomonotone operators and discrete Gronwall’s inequality to prove the existence of solutions and its priori estimates for the discrete approximation system. Furthermore, through a limiting procedure for solutions of the discrete approximation system, an existence theorem for the fractional evolution inclusion problem is established. Finally, as an illustrative application, a complicated quasistatic viscoelastic contact problem with a generalized Kelvin–Voigt constitutive law with fractional relaxation term and friction effect is considered.

A backward Euler difference scheme for the integro-differential equations with the multi-term kernels

ABSTRACTA backward Euler difference scheme is formulated and analyzed for the integro-differential equations with the multi-term kernels. The convolution quadrature is used to deal with the Riemann-Liouville fractional integral terms. A fully discrete difference scheme is constructed with time derivative by backward Euler scheme and space discretization by standard central difference approximation. We prove the stability and convergence based on the nonnegative character of the real quadratic form. The and -norms stability and convergence are given. Finally, numerical experiments validate the theoretical results.

A class of differential hemivariational inequalities in Banach spaces

In this paper we investigate an abstract system which consists of a hemivariational inequality of parabolic type combined with a nonlinear evolution equation in the framework of an evolution triple of spaces which is called a differential hemivariational inequality [(DHVI), for short]. A hybrid iterative system corresponding to (DHVI) is introduced by using a temporally semi-discrete method based on the backward Euler difference scheme, i.e., the Rothe method, and a feedback iterative technique. We apply a surjectivity result for pseudomonotone operators and properties of the Clarke subgradient operator to establish existence and a priori estimates for solutions to an approximate problem. Finally, through a limiting procedure for solutions of the hybrid iterative system, the solvability of (DHVI) is proved without imposing any convexity condition on the nonlinear function umapsto f(t,x,u) and compactness of C_0-semigroup e^{A(t)}.

Fourth-order central compact scheme for the numerical solution of incompressible Navier–Stokes equations

ABSTRACTThis paper provides an implicit central compact scheme for the numerical solution of incompressible Navier–Stokes equations. The solution procedure is based on the artificial compressibility method that transforms the governing equations into a hyperbolic-parabolic form. A fourth-order central compact scheme with a sixth-order numerical filtering is used for the discretization of convective terms and fourth-order central compact scheme for the viscous terms. Dual-time stepping approach is applied to time discretization with backward Euler difference scheme to the pseudo-time derivative, and three point second-order backward difference scheme to the physical time derivative. An approximate factorization-based alternating direction implicit scheme is used to solve the resulting block tridiagonal system of equations. The accuracy and efficiency of the proposed numerical method is verified by simulating several two-dimensional steady and unsteady benchmark problems.

Numerical bifurcation of a delayed diffusive food‐limited model with Dirichlet boundary condition

This paper studies the Neimark–Sacker bifurcation of a diffusive food‐limited model with a finite delay and Dirichlet boundary condition by the backward Euler difference scheme, Crank‐Nicolson difference scheme, and nonstandard finite‐difference scheme. The existence of Neimark‐Sacker bifurcation at the equilibrium is obtained. Our results show that Crank‐Nicolson and nonstandard finite‐difference schemes are superior to the backward Euler difference scheme under the means of describing approximately the dynamics of the original system. Finally, numerical examples are provided to illustrate the analytical results. Copyright © 2015 John Wiley & Sons, Ltd.

An adaptive moving grid method for a system of singularly perturbed initial value problems

A system of first-order singularly perturbed initial value problems is considered. The system is discretized by a backward Euler difference scheme for which a priori error analysis in the maximum norm is constructed. It is shown from the a priori error bound that there exists a mesh with N subintervals that gives optimal error bound of O(N−1) which is robust with respect to the perturbation parameters. A partly heuristic argument based on a priori error analysis leads to a suitable monitor function. Based on an a posteriori error bound, a first-order rate of convergence, independent of all perturbation parameters, is established. A linear and a nonlinear examples are tested, and the numerical results are provided to demonstrate the effectiveness of our adaptive moving grid method.

Second order global uniformly convergent numerical method for a coupled system of singularly perturbed initial value problems

We consider a coupled system of singularly perturbed first order ordinary differential equations. The equations have small positive parameters of different magnitudes associated with them. A decomposition of the exact solution into smooth and singular components is constructed that is useful for the analysis of the numerical method. We apply the HODIE (High Order Difference approximation with Identity Expansions) technique to construct a second order finite difference scheme and combine this with standard backward Euler difference scheme in a special way on a piecewise-uniform Shishkin mesh to solve the system numerically. It is proved that the method is second order (up to a logarithmic factor) convergent in the maximum norm uniformly in both perturbation parameters. Numerical experiments are conducted to support the theoretical results.

The stability and convergence of two linearized finite difference schemes for the nonlinear epitaxial growth model

AbstractThe numerical simulation of the dynamics of the molecular beam epitaxy (MBE) growth is considered in this article. The governing equation is a nonlinear evolutionary equation that is of linear fourth order derivative term and nonlinear second order derivative term in space. The main purpose of this work is to construct and analyze two linearized finite difference schemes for solving the MBE model. The linearized backward Euler difference scheme and the linearized Crank‐Nicolson difference scheme are derived. The unique solvability, unconditional stability and convergence are proved. The linearized Euler scheme is convergent with the convergence order of O(τ + h2) and linearized Crank‐Nicolson scheme is convergent with the convergence order of O(τ2 + h2) in discrete L2‐norm, respectively. Numerical stability with respect to the initial conditions is also obtained for both schemes. Numerical experiments are carried out to demonstrate the theoretical analysis. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011

Parameter-robust numerical method for a system of singularly perturbed initial value problems

In this work we study a system of M(???2) first-order singularly perturbed ordinary differential equations with given initial conditions. The leading term of each equation is multiplied by a distinct small positive parameter, which induces overlapping layers. A maximum principle does not, in general, hold for this system. It is discretized using backward Euler difference scheme for which a general convergence result is derived that allows to establish nodal convergence of O(N ???1ln N) on the Shishkin mesh and O(N ???1) on the Bakhvalov mesh, where N is the number of mesh intervals and the convergence is robust in all of the parameters. Numerical experiments are performed to support the theoretical results.

Finite element approximation of a forward and backward anisotropic diffusion model in image denoising and form generalization

AbstractA new forward–backward anisotropic diffusion model is introduced. The two limit cases are the Perona‐Malik equation and the Total Variation flow model. A fully discrete finite element scheme is studied using C0‐piecewise linear elements in space and the backward Euler difference scheme in time. A priori estimates are proven. Numerical results in image denoising and form generalization are presented.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

Optimal Convergence for the Implicit Space‐Time Discretization of Parabolic Systems with p‐Structure

Parabolic systems with $p$-structure are considered on convex polyhedral domains under Dirichlet boundary conditions. A fully discrete scheme is studied using $C^0$-piecewise linear finite elements in space and the backward Euler difference scheme in time. A priori error estimates in quasi norms are proved, and optimal convergence rates are obtained.

Robust integration schemes for generalized viscoplasticity with internal-state variables

This paper is concerned with the development of a general framework for the implicit time-stepping integrators for the flow and evolution equations in complex viscoplastic models. The primary goal is to present a complete theoretical formulation, and to address in detail the algorithmic and numerical analysis aspects involved in its finite element implementation, as well as to critically assess the numerical performance of the developed schemes in a comprehensive set of test cases. On the theoretical side, we use the unconditionally stable, backward Euler difference scheme. Its mathematical structure is of sufficient generality to allow a unified treatment of different classes of viscoplastic models with internal variables. Two specific models of this type, which are representatives of the present state-of-the-art in metal viscoplasticity, are considered in the applications reported here; i.e., generalized viscoplasticity with complete potential structure fully associative (GVIPS) and non-associative (NAV) models. The matrix forms developed for both these models are directly applicable for both initially isotropic and anisotropic materials, in three-dimensions as well as subspace applications (i.e., plane stress/strain, axisymmetric, generalized plane stress in shells). On the computational side, issues related to efficiency and robustness are emphasized in developing the (local) iterative algorithm. In particular, closed-form expressions for residual vectors and (consistent) material tangent stiffness arrays are given explicitly for the GVIPS model, with the maximum matrix sizes ‘optimized’ to depend only on the number of independent stress components (but independent of the number of viscoplastic internal state parameters). Significant robustness of the local iterative solution is provided by complementing the basic Newton–Raphson scheme with a local line search strategy for convergence.

An Implicit integration scheme for generalized viscoplasticity with dynamic recovery

This paper is concerned with the development of a general implicit time-stepping integrator for the flow and evolution equations in a recent representative class of generalized viscoplastic models, involving both hardening and dynamic recovery mechanisms. To this end, the computational framework is developed on the basis of the unconditionally stable, backward Euler difference scheme. Its mathematical structure is of sufficient generality to allow a systematic treatment of several internal variables of the tensorial and scalar types. The matrix forms developed are directly applicable in general (three-dimensional) situations as well as subspace applications (i.e., plane stress/strain, axisymmetric, generalized plane stress in shells). The closed-form expressions for residual vectors and the algorithmic, (consistent) material tangent stiffness array are given explicitly, with the maximum matrix sizes “optimized” to depend only on the number of independent stress components, but not the number of internal state variables involved. Several numerical simulations are given to assess the performance of the developed schemes.