Exponential Time Differencing Runge Kutta Research Articles

Fourth-order exponential time differencing Runge–Kutta scheme and local meshless method to investigate unsteady diffusion–convection problems of anisotropic functionally graded materials

The current article is devoted to proposing a local meshfree method for simulating the unsteady diffusion–convection problems of anisotropic functionally graded materials (AFGM). The first stage of the proposed numerical algorithm is based on discretizing the spatial variables by utilizing the direct meshless local Petrov–Galerkin (DMLPG) method. Then a system of ODEs is extracted which is associated to the time variable. Furthermore, the fourth-order exponential time differencing Runge–Kutta method is applied to solve the system of ODEs. The main aim of this work is to introduce a flexible and low-cost numerical procedure to solve unsteady diffusion–convection problems of AFGM on the complicated geometry. The new numerical technique has admissible resultants to simulate the studied model. This numerical formulation is applied to several examples with complex domains to check its ability.

Quantifying and eliminating the time delay in stabilization exponential time differencing Runge–Kutta schemes for the Allen–Cahn equation

Although the stabilization technique is favorable in designing unconditionally energy stable or maximum-principle-preserving schemes for gradient flow systems, the induced time delay is intractable in computations. In this paper, we propose a class of delay-free stabilization schemes for the Allen–Cahn gradient flow system. Considering the Fourier pseudo-spectral spatial discretization for the Allen–Cahn equation with either the polynomial or the logarithmic potential, we establish a semi-discrete, mesh-dependent maximum principle by adopting a stabilization technique. To unconditionally preserve the mesh-dependent maximum principle and energy stability, we investigate a family of exponential time differencing Runge–Kutta (ETDRK) integrators up to the second-order. After reformulating the ETDRK schemes as a class of parametric Runge–Kutta integrators, we quantify the lagging effect brought by stabilization, and eliminate delayed convergence using a relaxation technique. The temporal error estimate of the relaxation ETDRK integrators in the maximum norm topology is analyzed under a fixed spatial mesh. Numerical experiments demonstrate the delay-free and structure-preserving properties of the proposed schemes.

Energy-decreasing second order exponential time differencing Runge–Kutta methods for Nonlocal Cahn–Hilliard equation

Due to the energetic variational structure inherent in the nonlocal phase field model, the energy of the system naturally decreases over time according to the Nonlocal Cahn–Hilliard (NCH) equation. Preserving this crucial property of energy dissipation is highly desirable in numerical solutions. However, most existing numerical methods can only ensure a decrease in some modified form of energy, rather than the original energy itself. In this paper, we propose the second-order exponential time differencing Runge–Kutta (ETD-RK2) methods for solving the NCH equation. We demonstrate that this method has the ability to unconditionally preserve the original energy dissipation. Numerical experiments are included to illustrate the accuracy and energy stability of the ETD-RK2 methods.

A family of structure-preserving exponential time differencing Runge–Kutta schemes for the viscous Cahn–Hilliard equation

We consider the numerical approximations of the viscous Cahn–Hilliard equation with either the Ginzburg–Landau polynomial potential or Flory–Huggins logarithmic potential. One challenge in solving such a fourth-order-in-space system is to develop accurate temporal discretization to preserve the energy stability, mass conservation, and maximum principle. We resolve this issue by developing a family of first- and second-order time marching schemes based on exponential time differencing Runge–Kutta (ETDRK) methods. We prove that the proposed schemes unconditionally preserve the three characteristics of the viscous Cahn–Hilliard equation. Because of the preservation of maximum principle, an error estimate in the infinity-norm is derived for the fully discrete system. Various numerical experiments are performed to verify the theoretical results and demonstrate the excellent performance of the proposed schemes.

Energy-decreasing exponential time differencing Runge–Kutta methods for phase-field models

Gradient flow models attract much attention these years. The energy naturally decreases along the direction of gradient flows. In order to preserve this property, various numerical schemes have been developed and among them, a very significant approach is the exponential time differencing Runge–Kutta method (ETDRK). In this paper we prove that the second order ETDRK (ETDRK2) scheme unconditionally preserves the energy dissipation law for a family of phase field models, such as the Allen–Cahn equation, the Cahn–Hilliard equation and the molecular beam epitaxy (MBE) model. As far as we know, this is the first work to show that a second-order linear scheme can guarantee the dissipation of the original energy unconditionally, instead of dissipation of modified energy for most existing works. Furthermore, we present some numerical simulations to demonstrate the accuracy and stability of the ETDRK2 scheme by applying the spectral method.

Fast high-order method for multi-dimensional space-fractional reaction–diffusion equations with general boundary conditions

To achieve the efficient and accurate long-time integration, we propose a fast and stable high-order numerical method for solving fractional-in-space reaction–diffusion equations. The proposed method is explicit in nature and utilizes the fourth-order compact finite difference scheme and matrix transfer technique (MTT) in space with FFT-based implementation. Time integration is done through the modified fourth-order exponential time differencing Runge–Kutta scheme. The linear stability analysis and various numerical experiments including two-dimensional (2D) Fitzhugh–Nagumo, Allen–Cahn, Gierer–Meinhardt, Gray–Scott and three-dimensional (3D) Schnakenberg models are presented to demonstrate the accuracy, efficiency, and stability of the proposed method.

Fourth-order time stepping methods with matrix transfer technique for space-fractional reaction-diffusion equations

In this paper, we develop two high-order methods for space-fractional reaction-diffusion equations. The schemes are based on fourth-order Matrix Transfer Technique (MTT) for spatial discretization and fourth-order Exponential Time Differencing Runge–Kutta (ETDRK) for temporal discretization. The ETDRK schemes are based on diagonal and sub-diagonal Padé approximations to the matrix exponential functions. It is observed that the A-stable scheme incurs unwanted oscillations due to high-frequency components present in the solution. These oscillations diminish as the order of the space-fractional derivative decreases. We propose a reliability constraint which is dependent on the order of the space-fractional derivative to avoid these oscillations. However, the L-stable scheme is oscillation-free for any time step. Partial fraction splitting technique is used to compose computationally efficient versions of the schemes. The amplification factor of the schemes is investigated by plotting their stability regions. Convergence analysis is performed on numerical experiments to demonstrate the fourth-order accuracy of the developed schemes in space and time. Numerical experiments made on multidimensional space-fractional reaction-diffusion equations show the reliability, stability, and efficiency of the schemes.

Turing pattern in the fractional Gierer–Meinhardt model**Project supported by the National Natural Science Foundation of China (Grant Nos. 61573008 and 61703290) and Natural Science Foundation of Liaoning Province, China (Grant No. 20180550996).

It is well-known that reaction–diffusion systems are used to describe the pattern formation models. In this paper, we will investigate the pattern formation generated by the fractional reaction–diffusion systems. We first explore the mathematical mechanism of the pattern by applying the linear stability analysis for the fractional Gierer–Meinhardt system. Then, an efficient high-precision numerical scheme is used in the numerical simulation. The proposed method is based on an exponential time differencing Runge–Kutta method in temporal direction and a Fourier spectral method in spatial direction. This method has the advantages of high precision, better stability, and less storage. Numerical simulations show that the system control parameters and fractional order exponent have decisive influence on the generation of patterns. Our numerical results verify our theoretical results.

A second-order efficient L-stable numerical method for space fractional reaction–diffusion equations

ABSTRACTIn this article we are proposing a strongly stable computationally efficient method in time to numerically solve reaction–diffusion equations with space fractional derivative. The Riesz space fractional derivative is discretized using the second-order fractional centred difference method. Time stepping scheme is based on second-order exponential time differencing Runge–Kutta method. Second-order positivity preserving Padé approximations is used to develop the proposed L-stable method. The computation efficiency of the method is significantly enhanced by using partial fractions splitting technique. The method is shown to be stable and reliable. Solution profiles as well as convergence tables in time are presented for various values of diffusion rates α.

Fourth-order methods for space fractional reaction–diffusion equations with non-smooth data

ABSTRACTWe propose two fourth-order methods in time for one-dimensional space fractional reaction–diffusion equations. The methods are based on fourth-order Exponential Time Differencing Runge–Kutta method. Padé approximations of matrix exponential functions are used to construct an L-stable and an A-stable method. Partial fraction splitting technique is applied to construct more reliable and computationally efficient versions of the methods. Solution profiles as well as convergence rates in time are presented for fractional enzyme kinetics equation and fractional Fisher equation. The L-stable method performs well in the presence of non-smooth mismatched initial-boundary data while the A-stable method is more economical for smooth matched initial-boundary data.

The space-splitting idea combined with local radial basis function meshless approach to simulate conservation laws equations

One acceptable technique in meshfree methods is collocation procedure based on the radial basis functions. But the mentioned technique is poor for solving problems that have shock (advection problems). The use of local collocation radial basis function method overcomes this important issue. The interpolation matrix of local radial basis functions is well-posed. Also, it is better, for obtaining high-order accurate results, we consider many points in the domain; thus, the size of coefficient matrix will be large. We suggest a robust technique to deal with this matter. In the current paper, we combine the local collocation radial basis functions method with the space-splitting technique. Solving some one-dimensional problems instead of a two-dimensional problem is an important advantage of the presented combination. For showing the efficiency of new method, we solve some conservation laws equations. An important benefit of the meshless methods is applying them on irregular domains; thus, we provided a test problem that has been solved on two irregular geometries. We use the local RBF collocation method to discrete the spatial domain and then we obtain a nonlinear system of ODEs that it will be solved using the fourth-order exponential time differencing Runge–Kutta method. Moreover, several examples are given that show the acceptable accuracy and proficiency of the present scheme.

Numerical investigation based on direct meshless local Petrov Galerkin (direct MLPG) method for solving generalized Zakharov system in one and two dimensions and generalized Gross–Pitaevskii equation

In the current investigation, we propose two efficient meshless numerical techniques for solving two models in optic engineering, i.e. the generalized Gross–Pitaevskii equation and the generalized Zakharov system. Two local meshless methods have been employed for solving these models: local radial basis functions collocation method and direct meshless local Petrov–Galerkin method. In this paper, we discrete the space direction using the local radial basis functions collocation and direct meshless local Petrov–Galerkin techniques and to obtain high-order numerical results, we use the fourth-order exponential time differencing Runge–Kutta method for discretizing the temporal direction. The obtained numerical results are compared with some well-known numerical techniques. Moreover, several examples are given that show the acceptable accuracy and efficiency of the proposed scheme.

Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications

This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models.

Remediation of contaminated groundwater by meshless local weak forms

In this paper, four meshless local weak form methods such as direct meshless local Petrov–Galerkin method (DMLPG), meshless local Petrov–Galerkin method (MLPG), local weak radial point interpolation method (LWRPIM) and local weak moving kriging method (LWMK) are applied to find the numerical solution of coupled non-linear advection–diffusion–reaction system arising in the prevention of groundwater contamination problem. A comparison between these methods is done from the perspective of accuracy and computational efficiency. An efficient fourth-order exponential time differencing Runge–Kutta method is utilized for the time discretization. The computational efficiency is the most significant advantage of the DMLPG method in comparison with the other meshless local weak form methods, because DMLPG reduces the computational costs, significantly. This is due to the fact that DMLPG shifts the numerical integrations over low-degree polynomials rather than over complicated shape functions. The main aim of this paper is to show that the meshless local weak form methods can be used for solving the system of non-linear partial differential equations especially coupled non-linear advection–diffusion–reaction system. The numerical results confirm the good efficiency of the proposed methods for solving our model.

Numerical study of three-dimensional Turing patterns using a meshless method based on moving Kriging element free Galerkin (EFG) approach

In this paper a numerical procedure is presented for solving a class of three-dimensional Turing system. First, we discrete the spatial direction using element free Galerkin (EFG) method based on the shape functions of moving Kriging interpolation. Then, to achieve a high-order accuracy, we use the fourth-order exponential time differencing Runge–Kutta (ETDRK4) method. Using this discretization for the temporal dimension, we obtain an explicit scheme and do not need to solve nonlinear system of equations. The EFG method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the EFG method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Also building shape functions of MLS approximation is a time consuming procedure. Because of the mentioned reasons we employ the shape functions of moving Kriging interpolation technique which have the mentioned property and less CPU time is required for building them. For testing this method on three-dimensional PDEs, we select some equations and system of PDEs such as Allen–Cahn, Gray–Scott, Ginzburg–Landau, Brusselator models, predator–prey model with additional food supply to predator. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme.

Efficient and stable exponential time differencing Runge–Kutta methods for phase field elastic bending energy models

The Willmore flow formulated by phase field dynamics based on the elastic bending energy model has been widely used to describe the shape transformation of biological lipid vesicles. In this paper, we develop and investigate some efficient and stable numerical methods for simulating the unconstrained phase field Willmore dynamics and the phase field Willmore dynamics with fixed volume and surface area constraints. The proposed methods can be high-order accurate and are completely explicit in nature, by combining exponential time differencing Runge–Kutta approximations for time integration with spectral discretizations for spatial operators on regular meshes. We also incorporate novel linear operator splitting techniques into the numerical schemes to improve the discrete energy stability. In order to avoid extra numerical instability brought by use of large penalty parameters in solving the constrained phase field Willmore dynamics problem, a modified augmented Lagrange multiplier approach is proposed and adopted. Various numerical experiments are performed to demonstrate accuracy and stability of the proposed methods.

A compact fourth-order [formula omitted]-stable scheme for reaction–diffusion systems with nonsmooth data

Exponential time differencing Runge–Kutta (ETDRK) schemes based on diagonal Padé approximations for the numerical solution of reaction–diffusion systems containing nonsmooth data have the disadvantage of producing poor numerical results when the time steps are “too large” relative to the spatial steps. On the other hand, methods based on sub-diagonal Padé approximations do not suffer from this bottleneck. In this paper, we present a novel modified version of Cox and Matthews fourth-order ETDRK scheme based on sub-diagonal Padé approximation to matrix exponential functions in combination with compact fourth-order finite difference scheme (in space) for direct integration of reaction–diffusion problems with nonsmooth data. For an efficient implementation of the scheme, a partial fraction splitting technique is utilized in which it is required to solve several backward Euler-type linear systems at each time step. Moreover, the design of the algorithm offers parallel implementation, so we implement the proposed algorithm in parallel on two processors utilizing MatlabMPI (a parallel, message passing version of Matlab) and obtain that the parallel version is computationally more efficient than the existing schemes considered in this paper. We investigate the amplification factor of the scheme and plot its boundaries of stability regions which give an indication of the stability of the scheme. Calculation of the local truncation error and an empirical convergence analysis demonstrates the fourth-order accuracy of the proposed scheme. Accuracy, computational efficiency, and reliability of the new scheme are demonstrated with numerical examples and comparing it with existing schemes.

O33] Application of compound-energy formalism to the parabolic growth of solid metal oxides

Exponential time differencing Runge–Kutta (ETDRK) schemes based on diagonal Padé approximations for the numerical solution of reaction–diffusion systems containing nonsmooth data have the disadvantage of producing poor numerical results when the time steps are “too large” relative to the spatial steps. On the other hand, methods based on sub-diagonal Padé approximations do not suffer from this bottleneck. In this paper, we present a novel modified version of Cox and Matthews fourth-order ETDRK scheme based on sub-diagonal Padé approximation to matrix exponential functions in combination with compact fourth-order finite difference scheme (in space) for direct integration of reaction–diffusion problems with nonsmooth data. For an efficient implementation of the scheme, a partial fraction splitting technique is utilized in which it is required to solve several backward Euler-type linear systems at each time step. Moreover, the design of the algorithm offers parallel implementation, so we implement the proposed algorithm in parallel on two processors utilizing MatlabMPI (a parallel, message passing version of Matlab) and obtain that the parallel version is computationally more efficient than the existing schemes considered in this paper. We investigate the amplification factor of the scheme and plot its boundaries of stability regions which give an indication of the stability of the scheme. Calculation of the local truncation error and an empirical convergence analysis demonstrates the fourth-order accuracy of the proposed scheme. Accuracy, computational efficiency, and reliability of the new scheme are demonstrated with numerical examples and comparing it with existing schemes.

Fourth-order compact schemes for the numerical simulation of coupled Burgers’ equation

This paper introduces two new modified fourth-order exponential time differencing Runge–Kutta (ETDRK) schemes in combination with a global fourth-order compact finite difference scheme (in space) for direct integration of nonlinear coupled viscous Burgers’ equations in their original form without using any transformations or linearization techniques. One scheme is a modification of the Cox and Matthews ETDRK4 scheme based on (1,3)-Padé approximation and other is a modification of Krogstad’s ETDRK4-B scheme based on (2,2)-Padé approximation. Efficient versions of the proposed schemes are obtained by using a partial fraction splitting technique of rational functions. The stability properties of the proposed schemes are studied by plotting the stability regions, which provide an explanation of their behavior for dispersive and dissipative problems. The order of convergence of the schemes is examined empirically and found that the modification of ETDRK4 converges with the expected rate even if the initial data are nonsmooth. On the other hand, modification of ETDRK4-B suffers with order reduction if the initial data are nonsmooth. Several numerical experiments are carried out in order to demonstrate the performance and adaptability of the proposed schemes. The numerical results indicate that the proposed schemes provide better accuracy than other schemes available in the literature. Moreover, the results show that the modification of ETDRK4 is reliable and yields more accurate results than modification of ETDRK4-B, while solving problems with nonsmooth data or with high Reynolds number.

Limitations and improvements of standard spectral methods for pricing standard options

We discuss the issue of how to improve the efficiency of standard spectral methods for pricing options. As a prototype example, we consider American options which are frequently used in the market as they enjoyed the early exercise opportunities. Such options are often modeled by nonlinear partial differential equations, solutions of which are not obtainable in a closed form and thus necessitates robust numerical techniques. For the discretisation in space (asset) direction, we use a rational spectral methods. For time integration involved in the process, we looked at comparing the efficiency of the method by using contour integral method based on the inversion of the Laplace transform, and the Exponential Time Differencing Runge–Kutta method of order 4. It is known that when these spectral methods are applied to option pricing problems, they no longer retain their higher order convergence. This order reduction is attributed due to the non-smooth initial conditions. To overcome this, we propose a domain decomposition method based on our rational spectral method. For the problem under consideration, we divide the domain into two sub-domains, with the transition point as the strike price. To represent the solution in each sub-domains, we choose to approximate the solution by linear rational interpolants due to their improved stability properties as compared to the polynomial interpolant. Comparative results are also presented.