Exponential Time Differencing Research Articles

A fourth order Runge-Kutta type of exponential time differencing and triangular spectral element method for two dimensional nonlinear Maxwell's equations

In this paper, we study a numerical scheme to solve the nonlinear Maxwell's equations. The discrete scheme is based on the triangular spectral element method (TSEM) in space and the exponential time differencing fourth-order Runge-Kutta (ETDRK4) method in time. TSEM has the advantages of spectral accuracy and geometric flexibility. The ETD method involves exact integration of the linear part of the governing equation followed by an approximation of an integral involving the nonlinear terms. The RK4 scheme is introduced for the time integration of the nonlinear terms. The stability region of the ETDRK4 method is depicted. Moreover, the contour integral in the complex plan is utilized and improved to compute the matrix function required by the implementation of ETDRK4. The numerical results demonstrate that our proposed method is of exponential convergence with the order of basis function in space and fourth order accuracy in time.

Exponential time difference methods with spatial exponential approximations for solving boundary layer problems

AbstractIn this work, an efficient and stable exponential time difference method is presented for solving boundary layer problems. By combining exponential time difference schemes with spatial direct discontinuous Galerkin discretization based on exponential boundary layer approximations, the proposed algorithm not only may admit large time step sizes but also could provide good spatial approximations even if on rather coarse spatial grids in the boundary layer. Some energy stabilities of the numerical scheme are rigorously derived. Numerical examples illustrate the accuracy, stability and efficiency of the algorithm.

Energetic spectral-element time marching methods for phase-field nonlinear gradient systems

We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen–Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen–Cahn equation, we also apply the method to a conservative Allen–Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen–Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems.

Defining hybrid hierarchical models in πHYFLOW

Defining hybrid hierarchical models in <i>π</i>H<scp>Y</scp>F<scp>LOW</scp>

A Second-Order Exponential Time Differencing Multi-step Energy Stable Scheme for Swift–Hohenberg Equation with Quadratic–Cubic Nonlinear Term

A Second-Order Exponential Time Differencing Multi-step Energy Stable Scheme for Swift–Hohenberg Equation with Quadratic–Cubic Nonlinear Term

An ETD Method for Vulnerable American Options

This paper introduces the exponential time differencing (ETD) technique as a numerical method to efficiently solve vulnerable American options pricing. We address several challenges, including removing cross-derivative terms through appropriate transformations, treating early-exercise opportunities using the penalty method, and substituting fixed boundary conditions with corresponding one-sided finite differences. The proposed method is shown to be both accurate and efficient through numerical experiments, which also compare the results with existing methods and analyze the numerical stability and convergence rate.

Solution of The Duffing Equation Using Exponential Time Differencing Method

To describe the spring stiffening effect that occurs in physics and engineering problems, Georg Duffing added the cubic stiffness term to the linear harmonic oscillator equation and is now known as the Duffing oscillator. Despite its simplicity, its dynamic behavior is very diverse. In this research, the Exponential Time Difference method is introduced to solve the Duffing oscillator numerically. To formulate the ETD method, we were using the integration factors. It is a function which, when multiplied by an ordinary differential equation, produces a differential equation that can be integrated. This method is an effective numerical method for solving complex differential equations, especially equations that have strong non-linearity The ETD method delivers highly accurate numerical solutions for the Duffing oscillator, with minimal discrepancy from the analytical results. Through parameter variation, the ETD method's applicability extends to diverse Duffing oscillator configurations.

Exponential Time Differencing Scheme for Fractional Plasma Oscillations

Plasma represents one of the four basic states of matter, along with the solid, liquid, and gas states. Plasma oscillations represent a fundamental concept in plasma physics, and their understanding is crucial in various applications. Fractional order Lorentz models for plasma oscillations offer a more flexible framework for describing the complex dynamics and memory effects that are often observed in plasma systems. In this paper, we consider a fractional-order model for plasma oscillations. The model involves Caputo fractional derivative of order between one and two. An exponential time differencing method is developed to approximate the motion of electrons driven by zero, static, and oscillating electric fields.

Runge–Kutta type Time Stepping Methods for Space Fractional Reaction Diffusion Model with Restricted Padé Approximation

A new numerical method that solves a time-space-fractional reaction-diffusion equation effectively is presented. The space-fractional Riesz operator is first discretized using fourth order compact finite differences, resulting in a system with a linear stiff term. The system of differential equations is then solved through time integration using an exponential time difference method, which explicitly handles the nonlinear non-stiff term. Restricted Padé rational approximation of a single real pole is used to approximate the matrix exponential e-A. The problems concerning the stability and computational efficiency of this new approach are tackled by means of a splitting technique. Based on compact finite differences with third-order accuracy in time and restricted Padé approximation, this novel efficient technique reduces computing cost and effectively tackles the problems with non-smooth initial data. The superiority of new method in terms of accuracy, reliability, and computational efficiency is finally shown by numerical examples.

A hybrid fourth order time stepping method for space distributed order nonlinear reaction-diffusion equations

A new fourth order highly efficient Exponential Time Differencing Runge-Kutta type hybrid time stepping method is developed by hybriding two fourth order methods. The new hybrid method takes advantages of a positivity preserving L-stable method to damp unnatural oscillations due to low regularity of the initial data and of a computationally efficient A-stable method to achieve optimal order convergence. Computational efficiency and stability of the new method is further enhanced by applying a splitting technique which makes it possible to implement the method on parallel processors. A hybriding criteria is presented and an algorithm based on the hybrid method is developed. The method is implemented to solve two two-dimensional problems having Riesz space distributed order diffusion and nonlinear reaction terms, an Allen-Cahn equation with a cubic nonlinearity and an Enzyme Kinetics equation with a rational nonlinearity. Fourth-order temporal convergence is obtained through numerical experiments. A third test problem with exact solution available is also considered and both spatial as well as temporal orders of convergence are computed. High computational efficiency of the method is shown by recording the CPU time. Numerical solutions using different order strength distribution functions are also presented.

EXPONENTIAL-TIME DIFFERENCING METHOD FOR SOLVING REACTOR POINT KINETICS EQUATIONS WITH TEMPERATURE AND POWER FEEDBACK AND ITS VALIDATION BY ANALYSINGTHERMAL REACTOR BENCHMARKS

Exponential Time Differencing method with Taylor Series Expansion (ETD-TS) is used for solving nonlinear systems of point kinetics equations (PKEs). Integral forms of PKE are expanded using Taylor Series Expansion method. Time step value and Taylor series expansion order (n) is chosen to minimize the truncation error.PKEs are highly stiff and nonlinear in nature in presence of thermal feedback. So, advanced method (ETD-TS) is used in the present work to estimate the power transients. Thermal Reactors benchmark with stepandcompensated ramp reactivity insertions are considered for estimation of power transients with feedback. Comparison of results with other accurate numerical methods likeCATS (Convergence Accelerated Taylor Series), SCS (Shifted Chebyshev Series), ITS2 (Integral Taylor Series)etc confirms the validity and accuracy of this method.Time step of 10-2 – 10-4sec is considered for estimationof power and reactivity transients.

Optimal Error Estimates of the Semi-Discrete Local Discontinuous Galerkin Method and Exponential Time Differencing Schemes for the Thin Film Epitaxy Problem Without Slope Selection

Optimal Error Estimates of the Semi-Discrete Local Discontinuous Galerkin Method and Exponential Time Differencing Schemes for the Thin Film Epitaxy Problem Without Slope Selection

Unified Formulation of Leapfrog Complying Divergence Implicit FDTD Method for Lossy Media

The leapfrog alternating direction implicit finite-difference time-domain (ADI-FDTD) method for lossy media has been found to suffer from the non-physical field leakage problem in the non-penetrable structures, which gives rise to inaccuracies in modelling lossy media during scattering analyses. To address such problem, in this paper, we propose a unified formulation of leapfrog complying-divergence implicit (CDI)-FDTD method for lossy media without field leakage problem. Several different lossy schemes to incorporate the conductivity terms of lossy media, including exponential time differencing, averaging and forward-forward schemes are integrated into a unified formulation. The proposed leapfrog CDI-FDTD method not only resolves the problem of field leakage but also has complying divergence. Moreover, since the implicit update procedures in the proposed formulation are in the fundamental form with matrix-operator-free right-hand sides, the floating-point operations (flops) are fewer than that of the conventional leapfrog ADI-FDTD formulation for lossy media. Numerical results are given to demonstrate the advantages of the proposed method over the leapfrog ADI-FDTD method in accuracy, efficiency, complying divergence and resolving the field leakage problem for lossy media.

Energy stability of exponential time differencing schemes for the nonlocal Cahn‐Hilliard equation

AbstractThe nonlocal Cahn‐Hilliard equation has attracted much attention these years. Despite the advantage of describing more practical phenomena for modeling phase transitions of microstructures in materials, the nonlocal operator in the equation brings a lot of extra computational costs compared with the local Cahn‐Hilliard equation. Thus high order time integration schemes are needed in numerical simulations. In this paper, we propose two classes of exponential time differencing (ETD) schemes for solving the nonlocal Cahn‐Hilliard equation. We first use the Fourier collocation method to discretize the spatial domain, and then the ETD‐based multistep and Runge‐Kutta schemes are adopted for the time integration. In particular, some specific multistep and Runge‐Kutta schemes up to fourth order are constructed. We rigorously establish the energy stabilities of the multistep schemes up to fourth order and the second order Runge‐Kutta scheme, which show that the first order ETD and the second order Runge‐Kutta schemes unconditionally decrease the original energy. We also theoretically prove the mass conservations of the proposed schemes. Several numerical experiments in two and three dimensions are carried out to test the temporal convergence rates of the schemes and to verify their mass conservations and energy stabilities. The long time simulations of coarsening dynamics are also performed to verify the power law for the energy decay.

Local radial basis function collocation method preserving maximum and monotonicity principles for nonlinear differential equations

AbstractIn this paper, a hybrid numerical scheme based on combining exponential time differencing (ETD) and local radial basis function collocation method was constructed. Model problems with different boundary conditions were considered, and the resulting linear system was carefully analyzed. The relation between the number of points employed in the local radial basis function collocation method and the condition number of the coefficient matrix was given. For application, three typical differential equations were investigated, that is, the Allen–Cahn equation for checking the maximum‐preserving property, the combustion equation for checking the monotonicity‐preserving property, and the Gray–Scott system for checking the robustness of the proposed scheme. Numerical examples show the effectiveness of the proposed method.

Fully discrete error analysis of first‐order low regularity integrators for the Allen‐Cahn equation

AbstractThe Allen‐Cahn equation satisfies the maximum bound principle, that is, its solution is uniformly bounded for all time by a positive constant under appropriate initial and/or boundary conditions. It has been shown recently that the time‐discrete solutions produced by low regularity integrators (LRIs) are likewise bounded in the infinity norm; however, the corresponding fully discrete error analysis is still lacking. This work is concerned with convergence analysis of the fully discrete numerical solutions to the Allen‐Cahn equation obtained based on two first‐order LRIs in time and the central finite difference method in space. By utilizing some fundamental properties of the fully discrete system and the Duhamel's principle, we prove optimal error estimates of the numerical solutions in time and space while the exact solution is only assumed to be continuous in time. Numerical results are presented to confirm such error estimates and show that the solution obtained by the proposed LRI schemes is more accurate than the classical exponential time differencing (ETD) scheme of the same order.

An ETD method for multi‐asset American option pricing under jump‐diffusion model

In this paper, we propose a numerical method for American multi‐asset options under jump‐diffusion model based on the combination of the exponential time differencing (ETD) technique for the differential operator and Gauss–Hermite quadrature for the integral term. In order to simplify the computational stencil and improve characteristics of the ETD‐scheme mixed derivative eliminating transformation is applied. The results are compared with recently proposed methods.

Two efficient exponential energy-preserving schemes for the fractional Klein–Gordon Schrödinger equation

This paper constructs two efficient exponential energy-preserving schemes for solving the fractional Klein–Gordon Schrödinger equation. The developed schemes are built upon the newly proposed partitioned averaged vector field method and exponential time difference technique and enjoy some advantages of the partitioned averaged vector field method. In addition, the Fourier pseudo-spectral method is applied to discretize the fractional Laplacian operator to obtain schemes so that the FFT technique can be used to reduce the computational complexity of the developed schemes in long-time simulations. Finally, by solving some fractional Klein–Gordon Schrödinger equations, it is demonstrated that the proposed schemes are efficient, conserve energy, and have better numerical stability results than traditional schemes.

Energy landscape analysis for two-phase multi-component NVT flash systems by using ETD type high-index saddle dynamics

With the increasingly accurate modeling and simulation demands and techniques, obtaining the knowledge of the multi-component phase equilibrium states holds the bottleneck challenge in the study of multi-phase fluid flow. In order to resolve the shortages in computational efficiency and stability in the conventional iterative flash calculation, a new phase equilibrium prediction method is proposed by solving the saddle point of the multi-phase system. In this paper, the HiSD (High-index saddle point dynamics) algorithm is used for the first time to calculate the saddle points on the energy landscape of the two-phase two-component NVT flash model based on the Peng-Robinson equation of state, and the up-down search algorithm of HiSD is applied to generate the solution landscape of the flash system. The Rosenbrock-Euler ETD (exponential time differencing) format is involved to reduce the interference of system rigidity to the calculation. It can be referred from the numerical analysis that there are at most the 1st-order saddle points in the energy landscape of the two-component two-phase NVT flash system, and all these saddle points are located on one straight line of the hyperplane, where the energy is equal everywhere. All these 1st-order saddle points can converge to the same or equivalent local minima, which indicates that the two-component two-phase flash system is a system with only one single solution with physical meanings. In addition, the saddle points also obey a linear relationship and the energy remains the same at different temperatures. Therefore, using the method proposed in this paper, the conventional two-step efforts of phase stability test and phase separation calculation can be simplified. The 1st-order saddle points of the system can be directly calculated, reducing the need for an initial guess. The local minima can be directly searched through the downward direction of the saddle point, which greatly reduces the calculation amount of phase equilibrium calculations. Furthermore, the minimum states at different temperatures can be calculated in batch by using one certain initial value, which significantly improves the adaptability and reliability for complex engineering problems with drastic temperature changes.

A conservative exponential integrators method for fractional conservative differential equations

&lt;abstract&gt;&lt;p&gt;The paper constructs a conservative Fourier pseudo-spectral scheme for some conservative fractional partial differential equations. The scheme is obtained by using the exponential time difference averaged vector field method to approximate the time direction and applying the Fourier pseudo-spectral method to discretize the fractional Laplacian operator so that the FFT technique can be used to reduce the computational complexity in long-time simulations. In addition, the developed scheme can be applied to solve fractional Hamiltonian differential equations because the scheme constructed is built upon the general Hamiltonian form of the equations. The conservation and accuracy of the scheme are demonstrated by solving the fractional Schrödinger equation.&lt;/p&gt;&lt;/abstract&gt;