Explicit Euler Scheme Research Articles

Solution of a partial integro-differential equation arising from viscoelasticity

Partial integro-differential equations occur in many fields of science and engineering. This study presents various numerical schemes for solving a partial integro-differential equation with a weakly singular kernel. These schemes are presented for determining the solution of a partial integro-differential equation subject to an initial condition and given boundary conditions. We construct several computational techniques based on finite-difference schemes and the product trapezoidal numerical integration rule. This problem can be found in the modelling of physical phenomena involving viscoelasticity forces. The finite-difference procedures developed are based on the forward Euler explicit scheme, the backward Euler implicit technique, the Crank–Nicolson implicit formula and Crandall’s implicit method. Three of the methods have second-order accuracy with respect to the space variable. The order of accuracy of the Crandall’s scheme is higher than that of the others. The numerical results of a test problem are given to verify the stability and accuracy of the procedures discussed.

Numerical analysis of a modified finite element nonlinear Galerkin method

A fully discrete modified finite element nonlinear Galerkin method is presented for the two-dimensional equation of Navier-Stokes type. The spatial discretization is based on two finite element spaces XH and Xh defined on a coarse grid with grid size H and a fine grid with grid size h << H, respectively; the time discretization is based on the Euler explicit scheme with respect to the nonlinear term. We analyze the stability and convergence rate of the method. Comparing with the standard finite element Galerkin method and the nonlinear Galerkin method, this method can admit a larger time step under the same convergence rate of same order. Hence this method can save a large amount of computational time. Finally, we provide some numerical tests on this method, the standard finite element Galerkin method, and the nonlinear Galerkin method, which are in a good agreement with the theoretical analysis.

On the use of boundary-integral equation methods for unsteady free-surface flows

The Mixed Eulerian-Lagrangian Methods (MEL) forfree-surface potential flows solved by boundary-integral equations (BIEs) is considered, and the diffusion and dispersion errors are studied in the discrete linearized problem. The diffusion error is the base for the stability analysis of the scheme; both the errors give indications on the accuracy of the numerical solution. The study is divided into two steps: comparison of the discrete dispersion relation with the analytical solution and coupling with different time-integration schemes. In particular, a stability analysis of the Runge-Kutta and Taylor-expansion schemes, previously not given in the literature, is addressed. It is shown that MEL methods based on first- and second-order explicit Runge-Kutta and Taylor-expansion schemes are unstable, regardless of the technique adopted to discretize the BIEs. Higher-order Runge-Kutta and Taylor-expansion schemes lead to conditionally stable methods. Known results for explicit, implicit and explicit-implicit Euler schemes are recovered by the present analysis. The theoretical predictions of the errors are confirmed for two different boundary-element techniques: a high-order panel method based on B-Splines to solve for the velocity potential and a spectrally-accurate method based on the Euler-McLaurin summation formula to solve directly for the velocity field.

Numerical analysis of semilinear stochastic evolution equations in Banach spaces

The solution of stochastic evolution equations generally relies on numerical computation. Here, usually the main idea is to discretize the SPDE spatially obtaining a system of SDEs that can be solved by e.g., the Euler scheme. In this paper, we investigate the discretization error of semilinear stochastic evolution equations in L p -spaces, resp. Banach spaces. The space discretization may be done by Galerkin approximation, for the time discretization we consider the implicit Euler, the explicit Euler scheme and the Crank–Nicholson scheme. In the last section, we give some examples, i.e., we consider an SPDEs driven by nuclear Wiener noise approximated by wavelets and delay equation approximated by finite differences.

Numerical Analysis of Dusty Hypersonic Viscous Gas Flow over a Flat Plate

We present an analysis of dilute gas-particle flows at hypersonic Mach number in which the flow parameters are obtained by solving complete two-dimensional Navier-Stokes equations numerically for a Bat-plate geometry. The conservation equations for both the gas and particle phases have been solved using finite volume upwind schemes, Time marching is carried out using the Euler-implicit scheme for the gas phase and the Euler-explicit scheme for the particle phase. Roe's flux-difference-splitting scheme is used for the gas phase, and the Steger-Warming Bu-vector-splitting scheme is applied for the particle phase. The detailed Bow structures of the gas and particle phases are given in three distinct regions: the large-slip region near the leading edge, the moderate-slip region, and the small-slip region far downstream. It is found that the presence of solid particles in the flow has an appreciable effect on the flowfield parameters. Also the presence of particles enhance the skin friction and heal transfer along the wall and decreases the boundary-layer thickness compared with the pure-gas case.

Detection of the onset of numerical chaotic instabilities by lyapunov exponents

It is commonly found in the fixed‐step numerical integration of nonlinear differential equations that the size of the integration step is opposite related to the numerical stability of the scheme and to the speed of computation. We present a procedure that establishes a criterion to select the largest possible step size before the onset of chaotic numerical instabilities, based upon the observation that computational chaos does not occur in a smooth, continuous way, but rather abruptly, as detected by examining the largest Lyapunov exponent as a function of the step size. For completeness, examination of the bifurcation diagrams with the step reveals the complexity imposed by the algorithmic discretization, showing the robustness of a scheme to numerical instabilities, illustrated here for explicit and implicit Euler schemes. An example of numerical suppression of chaos is also provided.

Numerical modelling of thermal ablation phenomena due to a cathodicspot

A numerical simulation of the ablation problem for the cathode spotthat is based on the enthalpy formulation is presented and solved with afinite-element method using the Euler explicit scheme. Vaporization latentheat and ablation phenomena constitute the main difficulties of the cathodicsurface erosion. We deduce characteristic information related to the cathodespot such as the energy repartition in three phases and the ablation lengthfor different energy fluxes.

Dispersive Particle Transport: Identification of Macroscale Behavior in Heterogeneous Stratified Subsurface Flows

Dispersive mass transport processes in naturally heterogeneous geological formations (porous media) are investigated based on a particle approach to mass transport and on its numerical implementation using LPT3D, a Lagrangian Particle Tracking 3D code. We are currently using this approach for studying microscale and macroscale space-time behavior (advection, diffusion, dispersion) of tracer plumes, solutes, or miscible fluids, in 1,2,3-dimensional heterogeneous and anisotropic subsurface formations (aquifers, petroleum reservoirs). Our analyses are based on a general advection-diffusion model and numerical scheme where concentrations and fluxes are discretized in terms of particles. The advection-diffusion theory is presented in a probabilistic framework, and in particular, a numerical analysis is developed for the case of advective transport and rotational flows (numerical stability of the explicit Euler scheme). The remainder of the paper is devoted to the behavior of concentration, mass flux density, and statistical moments of the transported tracer plume in the case of heterogeneous steady flow fields, where macroscale dispersion occurs due to geologic heterogeneity and stratification. We focus on the case of perfectly stratified or multilayered media, obtained by generating many horizontal layers with a purely random transverse distribution of permeability and horizontal velocity. In this case, we calculate explicitly the exact mass concentration field C(x, t), mass flux density field f(x, t), and moments. This includes spatial moments and dispersion variance σ 2 x (t) on a finite domain L, and temporal moments on a finite time scale T, e.g., the mass variance of arrival times σ 2 T (x). The moments are related to flux concentrations in a way that takes explicitly into account finite space-time scales of analysis (time-dependent tracer mass; spatially variable flow through' mass). The multilayered model problem is then used in numerical experiments for testing different ways of recovering information on tracer plume migration, dispersion, concentration and flux fields. Our analyses rely on a probabilistic interpretation that emerges naturally from the particle approach; it is based on spatial moments (particle positions), temporal moments (mass weighted arrival times), and probability densities (both concentrations and fluxes). Finally, as an alternative to direct estimations of the flux and concentration fields, we formulate and study the Moment Inverse Problem. Solving the MIP yields an indirect method for estimating the space-time distribution of flux concentrations based on observed or estimated moments of the plume. The moments may be estimated from field measurements, or numerically computed by particle tracking as we do here.

A particle method for some parabolic equations

We present a quasi-Monte-Carlo particle simulation of some multidimensional linear parabolic equations with constant coefficients. We approximate the elliptic operator in space by a finite-difference operator. We discretize time into intervals of length Δt. The discrete representation of the solution at time t n = nΔt is a sum of Dirac delta measures. Using the explicit Euler scheme, the resulting approximation at time t n+1 is recovered by a quasi-Monte-Carlo integration. We make use of a technique involving renumbering the simulated particles in every time step. We state and prove a convergence theorem for the method. Experimental results are presented for some model problems. The results suggest that the quasi-Monte-Carlo simulation tends to give more accurate solutions than a Monte-Carlo simulation, when the correct renumbering technique is used. Other choices can result in significant loss of efficiency.

Convergence and stability of finite element nonlinear Galerkin method for the Navier-Stokes equations

In this article, we present a new fully discrete finite element nonlinear Galerkin method, which are well suited to the long time integration of the Navier-Stokes equations. Spatial discretization is based on two-grid finite element technique; time discretization is based on Euler explicit scheme with variable time step size. Moreover, we analyse the boundedness, convergence and stability condition of the finite element nonlinear Galerkin method. Our discussion shows that the time step constraints of the method depend only on the coarse grid parameter $H$ and the time step constraints of the finite element Galerkin method depend on the fine grid parameter $h « H$ under the same convergence accuracy.

Formulating Dynamic Multi-Rigid-Body Contact Problems with Friction as Solvable Linear Complementarity Problems

A linear complementarity formulation for dynamic multi-rigid-body contact problems with Coulomb friction is presented. The formulation, based on explicit Euler integration and polygonal approximation of the friction cone, is guaranteed to have a solution for any number of contacts and contact configuration. A model with the same property, based on the Poisson hypothesis, is formulated for impact problems with friction and nonzero restitution coefficients. An explicit Euler scheme based on these formulations is presented and is proved to have uniformly bounded velocities as the stepsize tends to zero for the Newton–Euler formulation in body co-ordinates.

On quasi-Monte Carlo simulation of stochastic differential equations

In a number of problems of mathematical physics and other fields stochastic differential equations are used to model certain phenomena. Often the solution of those problems can be obtained as a functional of the solution of some specific stochastic differential equation. Then we may use the idea of weak approximation to carry out numerical simulation. We analyze some complexity issues for a class of linear stochastic differential equations (Langevin type), which can be given by \[ d X t = − α X t d t + β ( t ) d W t , X 0 := 0 , dX_{t}=-\alpha X_{t}dt+\beta (t)dW_{t}, \quad X_{0}:= 0, \] where α &gt; 0 \alpha &gt;0 and β : [ 0 , T ] → R \beta : [0,T]\to \mathbb {R} . It turns out that for a class of input data which are not more than Lipschitz continuous the explicit Euler scheme gives rise to an optimal (by order) numerical method. Then we study numerical phenomena which occur when switching from (real) Monte Carlo simulation to quasi–Monte Carlo simulation, which is the case when we carry out the simulation on computers. It will easily be seen that completely uniformly distributed sequences yield good substitutes for random variates, while not all uniformly distributed (mod 1) sequences are suited. In fact we provide necessary conditions on a sequence in order to serve quasi–Monte Carlo purposes. This condition is expressed in terms of the measure of well-distributions. Numerical examples complement the theoretical analysis.

Algorithms for Phase Field Computation of the Dendritic Operating State at Large Supercoolings

This paper deals with numerical solutions of the phase field model of solidification in two dimensions in the context of dendritic growth. Finite difference methods associated with vectorized algorithms are employed to solve the phase field equations. The temperature equation is solved by an alternating direction implicit scheme and the equation for the phase field by an explicit Euler scheme. We identify a region of parameter space, namely large supercoolings, where we can compute dendritic morphologies that display steady-state dendrite tip operating conditions that are independent of computational parameters.

An Optimal Control Analysis of Water Pollution Problem

This paper presents an optimal control analysis applied to water pollution control using a combination of the finite element method and optimal control theory. A numerical model of water pollution can be expressed in terms of the linear two-dimensional shallow water and convection diffusion equations. These equations can be solved by the explicit Euler scheme. The method presented by Sakawa-Shindo is effectively used to implement the control theory. Considering the closed water region such as a lake or a coastal sea, into which some river is flowing, it is shown that the pollutant level in this region can be controlled by controlling the inflow velocity of the river. This paper describes the water pollution control problem with the use of inequality constraints.

Product integration methods for solving a system of nonlinear Volterra integral equations

In this paper the technique of subtracting out singularities is used to derive explicit and implicit product Euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of Volterra integral equations with a weakly singular kernel. The convergence proofs of the numerical schemes are presented; these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global Lipschitz condition.

On a Nonlinear Volterra Integro‐differential Equation with a Weakly Singular Kernel

This paper deals with a nonlinear Volterra integro-differential equation by discussing its equivalent Volterra integral equation system. An equivalent system of nonlinear Volterra integral equations is set up, an explicit Euler scheme is given and the existence and uniqueness of the solution of the original problem are demonstrated on [0, ∞) through the existence and uniqueness of the solution of the Volterra integral equation system.

Stability of weak numerical schemes for stochastic differential equations

We consider numerical stability and convergence of weak schemes solving stochastic differential equations. A relatively strong notion of stability for a special type of test equations is proposed. These are stochastic differential equations with multiplicative noise. For explicit and implicit Euler schemes the regions of stability are also examined.

Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh

We study here the discretisation of the nonlinear hyperbolic equationu t +div(vf(u))=0 in ?3 × ?+, with given initial conditionu(.,0)=u 0(.) in ?2, wherev is a function from ?2 × ?+ to ?2 such that divv=0 andf is a given nondecreasing function from ? to ?. An explicit Euler scheme is used for the time discretisation of the equation, and a triangular mesh for the spatial discretisation. Under a usual stability condition, we prove the convergence of the solution given by an upstream finite volume scheme towards the unique entropy weak solution to the equation.

The Global Dynamics of Discrete Semilinear Parabolic Equations

A class of scalar semilinear parabolic equations possessing absorbing sets, a Lyapunov functional, and a global attractor are considered. The gradient structure of the problem implies that, provided all steady states are isolated, solutions approach a steady state as $t \to \infty $. The dynamical properties of various finite difference and finite element schemes for the equations are analysed. The existence of absorbing sets, bounded independently of the mesh size, is proved for the numerical methods. Discrete Lyapunov functions are constructed to show that, under appropriate conditions on the mesh parameters, numerical orbits approach steady state solutions as discrete time increases. However, it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented. Furthermore, it is also shown that the explicit Euler scheme introduces spurious solutions with period 2 in the timestep. As a result, the absorbing set is destroyed and there is initial data leading to blow up of the scheme, however small the mesh parameters are taken. To obtain stabilization to a steady state for this scheme, it is necessary to restrict the timestep in terms of the initial data and the space step. Implicit schemes are constructed for which absorbing sets and Lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step; both one-step and multistep (BDF) methods are studied.

FORGING OF A CONNECTING ROD: 3D FINITE ELEMENT CALCULATIONS

This paper presents the results of the simulation of the forging of a connecting rod. The calculation has been carried out by the code FORGE3 developed at the CEMEF laboratory. FORGE3 is a three‐dimensional finite element computer program that can simulate hot forging of industrial parts. The flow problem is solved using a thermomechanical analysis. The mechanical resolution and the thermal one are coupled by the way of the consistency K which is thermodependent, the plastic deformation in the volume of the material and the friction heat flux on the surface. The material behaviour is assumed to be incompressible and viscoplastic (Norton—Hoff law) with the associated friction law. The thermal resolution includes the case of non‐linear physical properties and boundary conditions. An explicit Euler scheme is used for the mechanical resolution and two‐step schemes for the thermal one. For the computation of other parameters, it is necessary to have a good approximation for the strain rate tensor. The Orkisz method has been used to determine the deviatoric stress tensor and p is calculated by an original smoothing method. The results show that it is possible to get good information on the flow and on the physical properties during forging of automotive parts. Comparisons have been made with experimental measurements with a reasonably good agreement.