Forward Euler Scheme Research Articles

Vessel Enhancing Diffusion

Recently, an implementation of the Vessel Enhancing Diffusion (VED) algorithm [3] using the Insight Toolkit (ITK) framework [2] has been proposed by Enquobahrie et al.[1]. In this paper we present an alternative implementation, for two reasons. First, in this implementation all the main functionality of the algorithm, including eigensystem, vesselness, tensor calculation and PDE discretization using a forward Euler scheme are now grouped together in one single class. Although this may come at the cost of code-reusability, it improves readibility and enables application specific code optimization. The second reason is the criterion of reproducibility. Source code, test environment and example data of the paper [3] are provided.

Automatic substepping integration of the subloading tij model with stress path dependent hardening

This paper discusses the numerical integration of the subloading tij model. This is an elastoplastic model with stress path dependent hardening, which can predict the behaviour of normally consolidated clays or loose sands, as well as of over-consolidated clays or dense sands, with a small number of material parameters. Three features distinguish the subloading tij model from the conventional ones: (a) the use of a modified stress space given by tensor t ij ; (b) the split of the plastic strain increments in two components leading to a stress path dependent hardening; and (c) the use of two yield surfaces (subloading yield surface and normal yield surface). This last feature is based on the concept of sub-yielding stress states and adds an extra internal strain-like hardening variable, related to the relative density state, which demands its own evolution law. The three characteristics above greatly improve the prediction capabilities of the model, with respect to those of the well-known Cam clay model, at the cost of only two additional parameters. Nonetheless, the numerical integration of the constitutive equations of subloading tij model is a bit challenging, mainly due to the stress path dependent hardening. In order to integrate the equations of subloading tij model in the same way as for any conventional model, the authors reformulated its equations in a simpler and direct manner. Here, these equations are integrated using multi-step explicit schemes, such as modified-Euler and Runge–Kutta–Dormand–Price, with automatic error control. Simple forward-Euler scheme is also used for the sake of comparison. The results show that the modified-Euler scheme is more accurate as well as faster than the other schemes analysed over a wide range of error tolerance. Besides, the automatic feature of these schemes is a great convenience for the users of numerical codes.

Geometric and potential driving formation and evolution of biomolecular surfaces

This paper presents new geometrical flow equations for the theoretical modeling of biomolecular surfaces in the context of multiscale implicit solvent models. To account for the local variations near the biomolecular surfaces due to interactions between solvent molecules, and between solvent and solute molecules, we propose potential driven geometric flows, which balance the intrinsic geometric forces that would occur for a surface separating two homogeneous materials with the potential forces induced by the atomic interactions. Stochastic geometric flows are introduced to account for the random fluctuation and dissipation in density and pressure near the solvent-solute interface. Physical properties, such as free energy minimization (area decreasing) and incompressibility (volume preserving), are realized by some of our geometric flow equations. The proposed approach for geometric and potential forces driving the formation and evolution of biological surfaces is illustrated by extensive numerical experiments and compared with established minimal molecular surfaces and molecular surfaces. Local modification of biomolecular surfaces is demonstrated with potential driven geometric flows. High order geometric flows are also considered and tested in the present work for surface generation. Biomolecular surfaces generated by these approaches are typically free of geometric singularities. As the speed of surface generation is crucial to implicit solvent model based molecular dynamics, four numerical algorithms, a semi-implicit scheme, a Crank-Nicolson scheme, and two alternating direction implicit (ADI) schemes, are constructed and tested. Being either stable or conditionally stable but admitting a large critical time step size, these schemes overcome the stability constraint of the earlier forward Euler scheme. Aided with the Thomas algorithm, one of the ADI schemes is found to be very efficient as it balances the speed and accuracy.

A time-accurate explicit multi-scale technique for gas dynamics

We present a new time-accurate algorithm for the explicit numerical integration of the compressible Euler equations of gas dynamics. This technique is based on the discrete-event simulation (DES) methodology for nonlinear flux-conservative PDEs [Y.A. Omelchenko, H. Karimabadi, Self-adaptive time integration of flux-conservative equations with sources, J. Comput. Phys. 216 (1) (2006) 179–194]. DES enables adaptive distribution of CPU resources in accordance with local time scales of the underlying numerical solution. It distinctly stands apart from multiple (local) time-stepping algorithms in that it requires neither selecting a global synchronization time step nor pre-determining a sequence of time-integration operations for individual parts of a heterogeneous numerical system. In this paper we extend the DES methodology in three important directions: (i) we apply DES to a system of coupled gas dynamics equations discretized via a central-upwind scheme [A. Kurganov, E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations, J. Comput. Phys. 160 (2000) 241–282; A. Kurganov, S. Noelle, G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations, SIAM J. Sci. Comput. 23 (3) (2001) 707–740]; (ii) we introduce a new Preemptive Event Processing (PEP) technique, which automatically enforces synchronous execution of events with sufficiently close update times; (iii) we significantly improve the accuracy of the previous algorithm [Y.A. Omelchenko, H. Karimabadi, Self-adaptive time integration of flux-conservative equations with sources, J. Comput. Phys. 216 (1) (2006) 179–194] by applying locally second-order-in-time flux-conserving corrections to the solution obtained with the forward Euler scheme. The performance of the new technique is demonstrated in a series of one-dimensional gas dynamics test problems by comparing numerical solutions obtained in event-driven and equivalent time-stepping simulations.

Nonlinear parabolic problems with Neumann-type boundary conditions and L^1-data

In this paper, we study existence, uniqueness and stability questions for the nonlinear parabolic equation: ∂u ∂t −� pu + α(u )= fi n]0 ,T [×Ω, with Neumann-type boundary conditions and initial data in L 1 .O ur approach is based essentially on the time discretization technique by Euler forward scheme. 2000 Subject Classifications: 35K05, 35J55, 35D65.

An unconditionally stable numerical method for the Luo–Rudy 1 model used in simulations of defibrillation

Numerical simulations of defibrillation using the Bidomain model coupled to a model of membrane kinetics represent a serious numerical challenge. This is because very high voltages close to defibrillation electrodes demand that extreme time step restrictions be placed on standard numerical schemes, e.g. the forward Euler scheme. A common solution to this problem is to modify the cell model by simple if-tests applied to several equations and rate functions. These changes are motivated by numerical problems rather than physiology, and should therefore be avoided whenever possible. The purpose of this paper is to present a numerical scheme that handles the original model without modifications and which is unconditionally stable for the Luo–Rudy phase 1 model. This also shows that the cell model is mathematically well-behaved, even in the presence of very high voltages. Our theoretical results are illustrated by numerical computations.

Cluster integration method in Lagrangian particle dynamics

An efficient and robust approach is proposed in order to conduct numerical simulations of collisional particle dynamics in the Lagrangian framework. Clusters of particles are made of particles that interact or may interact during the next global time-step. Potential collision partners are found by performing a test move, that follows the patterns of a hard-sphere model. The clusters are integrated separately and the collisional forces between particles are given by a soft-sphere collision model. However, the present approach also allows longer range inter-particle forces. The integration of the clusters can be done by any one-step ordinary differential equation solver, but for dilute particle systems, the variable step-size Runge–Kutta solvers as the Dormand and Prince scheme [J. Comput. Appl. Math. 6 (1980) 19] are superior. The cluster integration method is applied on sedimentation of 5000 particles in a two-dimensional box. A significant speed-up is achieved. Compared to a traditional discrete element method with the forward Euler scheme, a speed-up factor of three orders of magnitude in the dilute regime and two orders of magnitude in the dense regime were observed. As long as the particles are dilute, the Dormand and Prince scheme is ten times faster than the classical fourth-order Runge–Kutta solver with fixed step size.

Robust discretizations versus increase of the time step for the Lorenz system

When continuous systems are discretized, their solutions depend on the time step chosen a priori. Such solutions are not necessarily spurious in the sense that they can still correspond to a solution of the differential equations but with a displacement in the parameter space. Consequently, it is of great interest to obtain discrete equations which are robust even when the discretization time step is large. In this paper, different discretizations of the Lorenz system are discussed versus the values of the discretization time step. It is shown that the sets of difference equations proposed are more robust versus increases of the time step than conventional discretizations built with standard schemes such as the forward Euler, backward Euler, or centered finite difference schemes. The nonstandard schemes used here are Mickens' scheme and Monaco and Normand-Cyrot's scheme.

Semidiscretization for a nonlocal parabolic problem

A time discretization technique by Euler forward scheme is proposed to deal with a nonlocal parabolic problem. Existence and uniqueness of the approximate solution are proved.

Computational Analysis of Tumor Angiogenesis Patterns Using a Two-dimensional Model

Tumor angiogenesis was simulated using a two-dimensional computational model. The equation that governed angiogenesis comprised a tumor angiogenesis factor (TAF) conservation equation in time and space, which was solved numerically using the Galerkin finite element method. The time derivative in the equation was approximated by a forward Euler scheme. A stochastic process model was used to simulate vessel formation and vessel elongation towards a paracrine site, i.e., tumor-secreted basic fibroblast growth factor (bFGF). In this study, we assumed a two-dimensional model that represented a thin (1.0 mm) slice of the tumor. The growth of the tumor over time was modeled according to the dynamic value of bFGF secreted within the tumor. The data used for the model were based on a previously reported model of a brain tumor in which four distinct stages (multicellular spherical, first detectable lesion, diagnosis, and death of the virtual patient) were modeled. In our study, computation was not continued beyond the 'diagnosis' time point to avoid the computational complexity of analyzing numerous vascular branches. The numerical solutions revealed that no bFGF remained within the region in which vessels developed, owing to the uptake of bFGF by endothelial cells. Consequently, a sharp declining gradient of bFGF existed near the surface of the tumor. The vascular architecture developed numerous branches close to the tumor surface (the brush-border effect). Asymmetrical tumor growth was associated with a greater degree of branching at the tumor surface.

Remarks on the rational approximation of holomorphic semigroups with nonuniform partitions

Some convergence and stability results for rational approximations of holomorphic semigroups are presented. The approximations are related to time discrete schemes with nonuniform stepsize to a linear evolution equation of parabolic type in a Banach space. Our theorems cover not only the backward Euler but also the Crank-Nicolson and forward Euler schemes.

Difference equations versus differential equations, a possible equivalence for the Rössler system?

When a set of nonlinear differential equations is investigated, most of time there is no analytical solution and only numerical integration techniques can provide accurate numerical solutions. In a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which the time variable is discrete. In numerical investigations a fourth-order Runge–Kutta integration scheme is usually sufficient. Nevertheless, sometimes a set of difference equations may be required and, in this case, standard schemes like the forward Euler, backward Euler or central difference schemes are used. The major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small integration time steps. In some cases, it may be of interest to obtain difference equations with the same type of solutions as for the differential equations but with significantly large time steps. Nonstandard schemes as introduced by Mickens [Nonstandard Finite Difference Models of Differential Equations, World Scientific, 1994] allow to obtain more robust difference equations. In this paper, using such nonstandard scheme, we propose some difference equations as discrete analogues of the Rössler system for which it is shown that the dynamics is less dependent on the time step size than when a nonstandard scheme is used. In particular, it has been observed that the solutions to the discrete models are topologically equivalent to the solutions to the Rössler system as long as the time step is less than the threshold value associated with the Nyquist criterion.

Viscosity of finite difference lattice Boltzmann models

Two-dimensional finite difference lattice Boltzmann models for single-component fluids are discussed and the corresponding macroscopic equations for mass and momentum conservation are derived by performing a Chapman–Enskog expansion. In order to recover the correct mass equation, characteristic-based finite difference schemes should be associated with the forward Euler scheme for the time derivative, while the space centered and second-order upwind schemes should be associated to second-order schemes for the time derivative. In the incompressible limit, the characteristic based schemes lead to spurious numerical contributions to the apparent value of the kinematic viscosity in addition to the physical value that enters the Navier–Stokes equation. Formulae for these spurious numerical viscosities are in agreement with results of simulations for the decay of shear waves.

Explicit integration method with time step control for viscoplasticity and creep

Constitutive equations for viscoplasticity and creep are expressed in terms of stress components, temperature and internal variables. The resulting set of ordinary differential equations of the first order under the guise of the finite element method is integrated by the Euler forward scheme with automatic subincrementation. The time step length is set on the basis of a posteriori error estimate. Testing examples were computed, including benchmark problems introduced by Zienkiewicz and Cormeau. In every case a perfect match between the converged step length and the Cormeau formula for the critical time step was observed.

溝型チップブレーカ工具による切削過程の三次元有限要素法解析

This paper describes the simulation analysis of 3-dimensional cutting processes with chip breaker. The simulation method is based on dynamic-explicit elastic-plastic finite element method (FEM) and heat transfer FEM using forward Euler scheme, which are suitable for an FE model including a large number of nodes such as 3-dimensional problem. First, the turning processes with flat rake face tools have been simulated for the basic investigation into the effects of rake angle. In the results, the radius of chip up-curl, cutting force and temperature decrease, while the radius of chip side-curl increases with increasing rake angle. Secondly, the cutting process with the groove tool has been simulated. The results make it clear that the groove on the rake face decreases the cutting force and temperature, and it improves the machinability. In addition, the simulation results with changing the primary land width indicate that the difference of tool contact area with chip greatly influences the 3-dimensional chip shape.

On the method of modified equations. III. Numerical techniques based on the second equivalent equation for the Euler forward difference method

Direct-correction and asymptotic successive-correction methods based on the second equivalent equation are applied to the Euler forward explicit scheme. In direct-correction, the truncation error terms of the second equivalent equation which contain higher-order derivatives together with a starting procedure, are discretized by means of finite differences. Both explicit and implicit direct-correction schemes are presented and their stability regions are studied. The asymptotic successive-correction numerical technique developed in Part II of this series with a consistent starting procedure is applied to the second equivalent equation. Both all-backward and all-centered asymptotic successive-correction methods are presented. The numerical methods introduced in this paper are applied to autonomous and non-autonomous, scalar and systems of ordinary differential equations and compared with the results of second- and fourth-order accurate Runge–Kutta methods. It is shown that the fourth-order Runge–Kutta method is more accurate than the successive-correction techniques for large time steps due to the need for higher-order derivatives of the Euler solution; however, for sufficiently small time steps, but larger enough so that round-off errors are negligible, both methods have nearly the same accuracy.

Suppression of numerically induced chaos with nonstandard finite difference schemes

It has been previously shown that despite its simplicity, appropriate nonstandard schemes greatly improve or eliminate numerical instabilities. In this work we construct several standard and nonstandard finite-difference schemes to solve a system of three ordinary nonlinear differential equations that models photoconductivity in semiconductors and for which it has been shown that integration with a conventional fourth-order Runge-Kutta algorithm produces numerical-induced chaos. It was found that a simple nonstandard forward Euler scheme successfully eliminates these numerical instabilities. In order to help determine the best finite-difference scheme, it was found useful to test the local stability of the scheme by direct inspection of the eigenvalues dependent on the step size.

Long range corrections to mixture properties of inhomogeneous systems

The Euler forward difference scheme applied to derive expressions for the long range corrections (LRC) to configurational energy, normal pressure, surface tension, and chemical potential of single-component inhomogeneous systems is extended to these properties of mixtures. The obtained expressions contain two parts and are consistent in their forms with those for pure fluids. Analytical expressions are also obtained for the first part when the Lennard-Jones potential is adopted in the evaluation. Simulations are carried out for four model binary mixtures (ε11=ε22=1.0, ε12=0.75, σ11=σ12=σ22=1.0, and P*=0.068, at T*=1.15 and one composition; ε11=1.0, ε22=0.597, ε12=0.773, σ11=1.0, σ12=0.884, and σ22=0.768, at T*=0.928 and three compositions) and mixtures of argon and krypton with εAr/k=119.8 K, εKr/k=163.1 K, σAr=3.405 Å, and σKr=3.63 Å, at T=119.8 K and three compositions. Comparisons of simulated results with those reported in the literature verify the applicability of the extension.

Long range corrections to thermodynamic properties of inhomogeneous systems with planar interfaces

Expressions for the long range corrections to configurational energy, normal pressure, surface tension, and chemical potential of an inhomogeneous system with two planar interfaces are developed. Applying the Euler forward difference scheme to correlate the density at any height to that of a central spatial element separates the corrections into two parts: one relates directly to the central local density, and another is due to the density differences between the central local value and those around the central spatial element. An analytical expression is obtained for the first part when the Lennard-Jones potential function is adopted in the evaluation. Variations of these properties along the normal direction are illustrated in terms of the equilibrium density profiles obtained from Monte Carlo simulations performed at T*=0.90 and T*=1.15.

Error Estimates for Finite Difference Methods for a Wide-Angle “Parabolic” Equation

We consider a model initial and boundary value problem for a third-order partial differential equation (PDE), a wide-angle parabolic equation frequently used in underwater acoustics, with depth- and range-dependent coefficients in the presence of horizontal interfaces and dissipation. After commenting on the existence--uniqueness theory of solution of the equation, we discretize the problem by a second-order finite difference method of Crank--Nicolson type for which we prove stability and optimal-order error estimates in suitable discrete L2-, H1-, and maximum norms. We also prove, under certain conditions, that the forward Euler scheme is also stable and convergent for the problem at hand.