Semi-implicit Euler Scheme Research Articles

Error analysis of a fully discrete projection method for Cahn–Hilliard Inductionless MHD problems

This article investigates the fully discrete finite element approximation and error analysis for a diffuse interface model of the two-phase incompressible inductionless magnetohydrodynamics problem. This model consists of Cahn–Hilliard equations, Navier–Stokes equations and Poisson equations, which are nonlinearly coupled through convection, stresses, and Lorentz forces. To address this highly nonlinear and multi-physics system, we propose a fully discrete energy stable scheme with the finite element projection method for spatial discretization, in which the velocity and pressure are decoupled. The temporal discretization is a combination of the first-order Euler semi-implicit scheme and a convex splitting energy strategy. We show that the proposed scheme is mass-conservative, charge-conservative and unconditional energy stable. The error estimates for the phase variable, chemical potential, velocity, pressure, current density and electric potential are rigorously established. Finally, several three-dimensional numerical experiments are performed to illustrate the features of the proposed numerical method and verify the theoretical conclusions.

Strong convergence analysis of spectral fractional diffusion equation driven by Gaussian noise with Hurst parameter less than [formula omitted]

We propose and analyze an efficient time discretization for the spectral fractional stochastic partial differential equation with Hurst parameter less than 12. By using variable substitution, the original equation is transformed into a system of equations, which includes a partial differential equation and a stochastic integral equation. Then the time discretization is composed of two parts. We discretize the partial differential equation in time by using a semi-implicit Euler scheme. Integration by parts is used to obtain the approximation of stochastic integral. We show the optimal error estimates of the time discretization in the strong sense. Finally, we provide some numerical examples to verify these theoretical results.

Analysis of a semi-implicit and structure-preserving finite element method for the incompressible MHD equations with magnetic-current formulation

In this paper, we investigate a fully discrete finite element scheme for the incompressible magnetohydrodynamic (MHD) equations with magnetic-current formulation that was introduced in Hu et al. (2016). We discretize the system by the semi-implicit Euler scheme in time and a mixed finite element approach together with finite element exterior calculus in space. The resulting scheme enjoys the structure-preserving feature that it can always produce an exactly divergence-free magnetic induction on the discrete level. The unique solvability and unconditional stability of the scheme are also proved rigorously. By utilizing the energy argument, error estimates for the velocity, magnetic induction, current density and induced electric field are further established under the low regularity hypothesis for the exact solutions. Numerical results are provided to verify the theoretical analysis and to show the effectiveness of the proposed scheme.

Decoupled, linear, unconditionally energy stable and charge-conservative finite element method for an inductionless magnetohydrodynamic phase-field model

In this paper, we consider the numerical approximation for a diffuse interface model of the two-phase incompressible inductionless magnetohydrodynamics (MHD) problem. This model consists of Cahn–Hilliard equations, Navier–Stokes equations and Poisson equation. We propose a linear and decoupled finite element method to solve this highly nonlinear and multi-physics system. For the time variable, the discretization is a combination of the first order Euler semi-implicit scheme, several first-order stabilization terms and implicit–explicit treatments for coupling terms. For the space variables, we adopt the finite element discretization. Especially, we approximate the current density and electric potential by inf–sup stable face-volume mixed finite element pairs. With these techniques, the scheme only involves a sequence of decoupled linear equations to solve at each time step. We show that the scheme is provably mass-conservative, charge-conservative and unconditionally energy stable. Numerical experiments are performed to illustrate the properties, accuracy and efficiency of the proposed scheme.

New error analysis and recovery technique of a class of fully discrete finite element methods for the dynamical inductionless MHD equations

In this paper, we present a new error analysis of a class of fully discrete finite element methods for the dynamical inductionless magnetohydrodynamic (MHD) equations. The methods use the semi-implicit backward Euler scheme in time and use the standard inf–sup stable Mini/Taylor–Hood pairs to discretize the velocity and pressure, and the Raviart–Thomas for solving the current density in space. Due to the strong coupling of the system and the pollution of the lower-order Raviart–Thomas face approximation in analysis, the existing analysis is not optimal. In terms of a mixed Poisson projection and the corresponding estimates in negative norms, we establish new and optimal error estimates for all variables. In particular, we prove that the method with the lowest-order Raviart–Thomas face element and Mini element provides the optimal accuracy for the velocity in L∞(0,T;L2(Ω))-norm, and the method with the lowest-order Raviart–Thomas face element and P2−P1 Taylor–Hood element supplies the optimal accuracy for the velocity in L2(0,T;H1(Ω))-norm and the pressure in L1(0,T;L2(Ω))-norm. Moreover, we propose a simple recovery technique to obtain a new numerical current density of first-order higher accuracy in the spatial direction by re-solving a mixed Poisson equation. Numerical experiments are performed to verify the theoretical analysis.

An operator splitting scheme for numerical simulation of spinodal decomposition and microstructure evolution of binary alloys

This article compares the operator splitting scheme to linearly stabilized splitting and semi-implicit Euler's schemes for the numerical solution of the Cahn-Hilliard equation. For the purpose of validation, the spinodal decomposition phenomena have been simulated. The efficacy of the three schemes has been demonstrated through numerical experiments. The computed results show that the schemes are conditionally stable. It has been observed that the operator splitting scheme is computationally more efficient.

Unconditional Superconvergence Error Estimates of Semi-Implicit Low-Order Conforming Mixed Finite Element Method for Time-Dependent Navier–Stokes Equations

In this paper, the unconditional superconvergence error analysis of the semi-implicit Euler scheme with low-order conforming mixed finite element discretization is investigated for time-dependent Navier–Stokes equations. In terms of the high-accuracy error estimates of the low-order finite element pair on the rectangular mesh and the unconditional boundedness of the numerical solution in L∞-norm, the superclose error estimates for velocity in H1-norm and pressure in L2-norm are derived firstly by dealing with the trilinear term carefully and skillfully. Then, the global superconvergence results are obtained with the aid of the interpolation post-processing technique. Finally, some numerical experiments are carried out to support the theoretical findings.

Unconditional stability and convergence analysis of fully discrete stabilized finite volume method for the time-dependent incompressible MHD flow

In this paper, we consider the stability and convergence of fully discrete stabilized finite volume method for the unsteady incompressible magnetohydrodynamics (MHD) equations on the quasi-uniform and regular triangulations. The spatial discretization is based on the lowest equal-order mixed finite element pair for the velocity, pressure and the magnetic fields, while the time discretization is based on the backward Euler semi-implicit scheme. In order to overcome the restriction of the discrete inf-sup condition, the local pressure projection stabilized method on the element is employed. This stabilized method has the advantages of parameter-free, no need to calculate the high-order derivatives or the edge-based data structures and it can be implemented at the element level. The unconditional stability of numerical scheme is established by using the Gronwall lemma, mathematical induction and choosing different test functions. Optimal error estimates of numerical approximations in various norms are also presented by constructing the corresponding dual problem. Finally, some numerical results are proposed to verify the established theoretical findings and show the performances of the considered numerical method.

A decoupled, unconditionally energy stable and charge-conservative finite element method for inductionless magnetohydrodynamic equations

In this paper, we propose a decoupled, unconditionally energy stable and charge-conservative finite element method for the inductionless magnetohydrodynamic equations. The time marching is combined with a first order semi-implicit Euler scheme, a first order perturbation term and some delicate implicit-explicit treatments for coupling terms. The finite element discretization is based on mixed conforming elements, where the hydrodynamic unknowns are approximated by stable finite element pairs, and the current density and electric potential are discretized by divergence-conforming face elements and discontinuous volume elements. This fully discrete scheme only needs to solve a linear subsystem at each time step and yields an exactly divergence-free current density directly. We show that the proposed scheme is well-posed and unconditionally energy stable. Under a weak regularity hypothesis on the exact solution, we present the error estimates for velocity, current density and electric potential. Finally, the validity, reliability, and accuracy of the proposed algorithm are supported by several numerical experiments.

Numerical Integration Scheme for Coupled Elastoplastic–Viscoplastic Constitutive Law for Tunnels

The paper presents an efficient numerical integration scheme for coupled elastoplastic–viscoplastic constitutive behavior with internal-state variables standing for irreversible processes. In most quasi-static structural analyses, the solution to boundary value problems involving materials that exhibit time-dependent constitutive behavior proceeds from the equation integration, handled at two distinct levels. On the one hand, the first, or local, level refers to the numerical integration at each Gaussian point of the rate constitutive stress–strain relationships. For a given strain increment, the procedure of local integration is iterated for stresses and associated internal variables until convergence of the algorithm is achieved. On the other hand, the second, or global, level is related to structure equilibrium between internal and external forces achieved by the Newton–Raphson iterative scheme. A review of the elastoplastic and viscoplastic model is given, followed by the coupling between these models. Particular emphasis is given in this contribution to address the first level integration procedure, also referred to as the algorithm for stress and internal variable update, considering a general elastoplastic–viscoplastic constitutive behavior. The formulation is described for semi-implicit Euler schemes. The efficacy of the numerical formulation is assessed by comparison with analytical and numerical solutions derived for deep tunnels in coupled elastoplasticity–viscoplasticity. Finally, a parametric analysis is performed to show the importance that this model can have, in the long-term convergence profile, against other models. For the considered flow surfaces, potential functions, and properties, differences on the order of 23% to 52% are found in the long-term convergence profile.

Error Analysis of a PFEM Based on the Euler Semi-Implicit Scheme for the Unsteady MHD Equations

In this article, we mainly consider a first order penalty finite element method (PFEM) for the 2D/3D unsteady incompressible magnetohydrodynamic (MHD) equations. The penalty method applies a penalty term to relax the constraint “”, which allows us to transform the saddle point problem into two smaller problems to solve. The Euler semi-implicit scheme is based on a first order backward difference formula for time discretization and semi-implicit treatments for nonlinear terms. It is worth mentioning that the error estimates of the fully discrete PFEM are rigorously derived, which depend on the penalty parameter , the time-step size , and the mesh size h. Finally, two numerical tests show that our scheme is effective.

CMMSE: numerical analysis of a chemical targeting model

Treating specific tissues without affecting other regions is a difficult task. It is desirable to target the particular tissue where the chemical has its biological effect. To study this phenomenon computationally, in this work we numerically study a mathematical model which is written as a nonlinear system composed by three parabolic partial differential equations. The variables involved in the model are the concentration of the chemical, the concentration of the binding protein and the concentration of the chemical bound to the protein. Our aim is to propose a fully discrete approximation of this problem, using the Finite Element Method and a semi-implicit Euler scheme, in order to solve it numerically. This discrete problem is analysed, obtaining a discrete stability property and some a priori error estimates that show the algorithm converges linearly if the continuous solution is regular enough. Also, some representative examples are shown, as well as the numerical verification of the convergence.

Error analysis of a conservative finite element scheme for time-dependent inductionless MHD problem

In this paper, we develop and analyze a fully discrete mixed finite element method for unsteady inductionless magnetohydrodynamics problem. An Euler semi-implicit scheme is proposed with a mixed variational formulation based on the variables (u,p,j,ϕ), in which the Navier–Stokes equations are approximated by stable finite elements and the current density is discretized by the divergence-conforming finite element. This scheme has the feature that the discrete current density keeps charge conservation property. It is shown that both the continuous problem and its fully discrete Euler semi-implicit scheme are well-posed. We prove unconditionally convergence and error estimates for velocity, pressure and current density. Finally, numerical experiments have been performed to validate the theoretical analysis and the law of charge conservation.

Convergence analysis of a conservative finite element scheme for the thermally coupled incompressible inductionless MHD problem

In this paper, a stable mixed finite element method for unsteady inductionless magnetohydrodynamics (MHD) problem coupled heat equation is devised. We first propose a mixed variational formulation based on all the variables, in which the thermal equation and Navier-Stokes equations are approximated by mini finite element method and the current density is discretized by the divergence-conforming elements. The novel feature of this scheme is that the discrete current density keeps charge conservation property. It is shown that the fully discrete first order Euler semi-implicit scheme is well-posed and unconditionally stable. The existence and uniqueness of the weak solutions for continuous problem is established by a numerical version analysis. Furthermore, under the low regularity hypothesis for the exact solutions, we prove the optimal error estimates of the velocity, current density and electric potential. Finally, some numerical experiments have been performed to validate the theoretical analysis and the law of charge conservation.

Stability of Euler Methods for Fuzzy Differential Equation

The Liu process is a fuzzy process whose membership function is a symmetric function on an expected value. The object of this paper was a fuzzy differential equation driven by Liu process. Since the existing fuzzy Euler solving methods (explicit Euler scheme, semi-implicit Euler scheme, and implicit Euler scheme) have the same convergence, to compare them, we presented four stabilities, i.e., asymptotical stability, mean square stability, exponential stability, and A stability. By choosing special fuzzy differential equation as a test equation, we deduced that mean square stability is equivalent to exponential stability. Furthermore, an explicit fuzzy Euler scheme and semi-implicit fuzzy Euler scheme showed asymptotical stability and mean square stability, while an explicit fuzzy Euler scheme failed to meet A stability but that an implicit fuzzy Euler scheme is A stable, and whether semi-implicit fuzzy Euler scheme is A stable depends on the values of α and λ.

SEINA: A two-dimensional steam explosion integrated analysis code

In the event of a severe accident, the reactor core may melt due to insufficient cooling. the high-temperature core melt will have a strong interaction (FCI) with the coolant, which may lead to steam explosion. Steam explosion would pose a serious threat to the safety of the reactors. Therefore, the study of steam explosion is of great significance to the assessment of severe accidents in nuclear reactors.This research focuses on the development of a two-dimensional steam explosion integrated analysis code called SEINA. Based on the semi-implicit Euler scheme, the three-phase field was considered in this code. Besides, the influence of evaporation drag of melt and the influence of solidified shell during the process of melt droplet fragmentation were also considered. The code was simulated and validated by FARO L-14 and KROTOS KS-2 experiments. The calculation results of SEINA code are in good agreement with the experimental results, and the results show that if the effects of evaporation drag and melt solidification shell are considered, the FCI process can be described more accurately. Therefore, it is proved that SEINA has the potential to be a powerful and effective tool for the analysis of steam explosions in nuclear reactors.

Convergence analysis of a fully discrete finite element method for thermally coupled incompressible MHD problems with temperature-dependent coefficients

In this paper, we study a fully discrete finite element scheme of thermally coupled incompressible magnetohydrodynamic with temperature-dependent coefficients in Lipschitz domain. The variable coefficients in the MHD system and possible nonconvex domain may cause nonsmooth solutions. We propose a fully discrete Euler semi-implicit scheme with the magnetic equation approximated by Nédélec edge elements to capture the physical solutions. The fully discrete scheme only needs to solve one linear system at each time step and is unconditionally stable. Utilizing the stability of the numerical scheme and the compactness method, the existence of weak solution to the thermally coupled MHD model in three dimensions is established. Furthermore, the uniqueness of weak solution and the convergence of the proposed numerical method are also rigorously derived. Under the hypothesis of a low regularity for the exact solution, we rigorously establish the error estimates for the velocity, temperature and magnetic induction unconditionally in the sense that the time step is independent of the spacial mesh size.

Temporal error analysis of a new Euler semi-implicit scheme for the incompressible Navier–Stokes equations with variable density

Based upon an equivalent form of the incompressible Navier–Stokes equations with variable density, a first-order backward Euler time-discrete scheme is studied for solving the variable density flows numerically. The proposed numerical scheme is unconditionally stable. A rigorous error analysis is presented and the first-order temporal convergence rate O(τ) is proved without any assumption on numerical solution by using the discrete maximal Lp-regularity of the Stokes problem, where τ is the time step size. Some numerical results are provided to confirm our theoretical analysis.

High-Accuracy Time Discretization of Stochastic Fractional Diffusion Equation

A high-accuracy time discretization is discussed to numerically solve the nonlinear fractional diffusion equation forced by a space-time white noise. The main purpose of this paper is to improve the temporal convergence rate by modifying the semi-implicit Euler scheme. The solution of the equation is only Hölder continuous in time, which is disadvantageous to improve the temporal convergence rate. Firstly, the system is transformed into an equivalent form having better regularity than the original one in time. But the regularity of nonlinear term remains unchanged. Then, combining Lagrange mean value theorem and independent increments of Brownian motion leads to a higher accuracy discretization of nonlinear term which ensures the implementation of the proposed time discretization scheme without loss of convergence rate. Our scheme can improve the convergence rate from $${\min \{\frac{\gamma }{2\alpha },\frac{1}{2}\}}$$ to $${\min \{\frac{\gamma }{\alpha },1\}}$$ in the sense of mean-squared $$L^2$$ -norm. The theoretical error estimates are confirmed by extensive numerical experiments.

Temporal error analysis of Euler semi-implicit scheme for the magnetohydrodynamics equations with variable density

In this paper, we consider the magnetohydrodynamics (MHD) equations with variable density, which are a coupled system by the incompressible Navier-Stokes equations with variable density and the Maxwell equations. Such MHD system describes the motions of several conducting incompressible immiscible fluids without surface tension in presence of a magnetic field. A first-order Euler semi-implicit time discrete scheme is proposed to approximate the MHD system such that we only need to solve the linearized subproblems at the discrete level. Moreover, it is unconditionally stable which is a key issue for problems of multiphysical fields. A rigorous error analysis is presented and the first-order temporal convergence rate O(τ) is derived for small τ by using the discrete Lp-regularity technique, where τ is the time step. The numerical results are shown to confirm the unconditional stability and the convergence rate.