Exponential Runge-Kutta Research Articles

Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations

In this paper, we study the classical Allen-Cahn equations and investigate the maximum-principle-preserving (MPP) techniques. The Allen-Cahn equation has been widely used in mathematical models for problems in materials science and fluid dynamics. It enjoys the energy stability and the maximum-principle. Moreover, it is well known that the Allen-Cahn equation may yield thin interface layer, and nonuniform meshes might be useful in the numerical solutions. Therefore, we apply the local discontinuous Galerkin (LDG) method due to its flexibility on h-p adaptivity and complex geometry. However, the MPP LDG methods require slope limiters, then the energy stability may not be easy to obtain. In this paper, we only discuss the MPP technique and use numerical experiments to demonstrate the energy decay property. Moreover, due to the stiff source given in the equation, we use the conservative modified exponential Runge-Kutta methods and thus can use relatively large time step sizes. Thanks to the conservative time integration, the bounds of the unknown function will not decay. Numerical experiments will be given to demonstrate the good performance of the MPP LDG scheme.

Projected exponential Runge–Kutta methods for preserving dissipative properties of perturbed constrained Hamiltonian systems

Preserving conservative and dissipative properties of dynamical systems is desirable in numerical integration. To this end, we develop and implement numerical methods that preserve the exact rate of dissipation in certain qualitative properties of dissipatively perturbed constrained Hamiltonian systems, which are shown to be conformal symplectic. Projection methods based on exponential Runge–Kutta methods are proposed for such systems. These numerical schemes are shown to be constraint preserving, conformal invariants preserving, symmetric, and second-order accurate. It is shown that these structure-preserving methods can be further composed to obtain higher-order structure-preserving methods. Linear stability analysis is used to derive stability properties and conformal preservation of the phase-space area. Numerical experiments, including constrained oscillators and a damped Korteweg–de Vries partial differential equation, demonstrate the advantages of geometric integration and verify theoretical results.

Efficient exponential Runge–Kutta methods of high order: construction and implementation

Exponential Runge–Kutta methods have shown to be competitive for the time integration of stiff semilinear parabolic PDEs. The current construction of stiffly accurate exponential Runge–Kutta methods, however, relies on a convergence result that requires weakening many of the order conditions, resulting in schemes whose stages must be implemented in a sequential way. In this work, after showing a stronger convergence result, we are able to derive two new families of fourth- and fifth-order exponential Runge–Kutta methods, which, in contrast to the existing methods, have multiple stages that are independent of one another and share the same format, thereby allowing them to be implemented in parallel or simultaneously, and making the methods to behave like using with much less stages. Moreover, all of their stages involve only one linear combination of the product of \(\varphi \)-functions (using the same argument) with vectors. Overall, these features make these new methods to be much more efficient to implement when compared to the existing methods of the same orders. Numerical experiments on a one-dimensional semilinear parabolic problem, a nonlinear Schrödinger equation, and a two-dimensional Gray–Scott model are given to confirm the accuracy and efficiency of the two newly constructed methods.

High order explicit exponential Runge–Kutta methods for semilinear delay differential equations

This paper aims to analyze order conditions of high order explicit exponential Runge–Kutta methods for stiff semilinear delay differential equations. Under the framework of analytic semigroup and the natural assumptions on the delay differential equations, the stiff order conditions up to order five are derived. Further, we show the method is stiffly convergent of order p even if the order conditions of order p holding in a weak form. Numerical tests are carried out to demonstrate the superiority of high order methods.

The fidelity of exponential and IMEX integrators for wave turbulence: Introduction of a new near-minimax integrating factor scheme

Exponential and IMEX integrators are considered for wave turbulence problems, which are stiff systems that often require long-time simulations. Applying a multi-scale asymptotic analysis to the difference equations for a simple wave turbulence problem reveals that Exponential Runge-Kutta methods incorrectly model interactions between resonant waves unless they are specifically designed otherwise, and IMplicit-EXplicit (IMEX) integrators introduce dispersive and dissipative errors that severely degrade the accuracy of slow energy exchange between resonant waves. The same analysis shows Integrating Factor (IF) methods to be the most suitable. Numerical experiments performed on the resonant triad model and the Majda-McLaughlin-Tabak (MMT) model verify the discrete-asymptotic analysis. In the MMT problem, Exponential Time Differencing methods require a time step more than two orders of magnitude smaller than IF methods to achieve comparable accuracy, and IMEX methods require more than six orders of magnitude smaller time steps. A novel near-minimax rational approximation of the matrix exponential is proposed to be used for integrating factor methods, and is demonstrated to be as accurate as the IF method but without the need to compute a matrix exponential.

Mass- and energy-preserving exponential Runge–Kutta methods for the nonlinear Schrödinger equation

In this paper, a family of arbitrarily high-order structure-preserving exponential Runge–Kutta methods are developed for the nonlinear Schrödinger equation by combining the scalar auxiliary variable approach with the exponential Runge–Kutta method. By introducing an auxiliary variable, we first transform the original model into an equivalent system which admits both mass and modified energy conservation laws. Then applying the Lawson method and the symplectic Runge–Kutta method in time, we derive a class of mass- and energy-preserving time-discrete schemes which are arbitrarily high-order in time. Numerical experiments are addressed to demonstrate the accuracy and effectiveness of the newly proposed schemes.

Exponential integrators for linear inhomogeneous problems

We consider the mildly stiff and stiff inhomogeneous linear initial value Problems sharing constant coefficients. Exponential Runge–Kutta methods are considered to tackle this problem. For this type of problem, we were able to save a function evaluation (stage) per step compared to the best available methods. This is important, as seen in various computational experiments where our current approach outperforms older ones.

Explicit exponential Runge–Kutta methods for semilinear parabolic delay differential equations

This paper is concerned with explicit exponential Runge–Kutta methods for semilinear parabolic delay differential equations. Stiff convergence and conditional DN-stability of explicit exponential Runge–Kutta methods are investigated in the framework of analytic semigroup on a Banach space. We derive the stiff convergence order conditions up to order four. In particular, it is shown that explicit exponential Runge–Kutta methods are conditionally DN-stable. Finally, numerical experiments are presented to validate the convergence results.

Exponential Runge–Kutta Method for Two-Dimensional Nonlinear Fractional Complex Ginzburg–Landau Equations

In this work, we study numerically two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equations. A centered finite difference method is exploited to discretize the spatial variables and leads to a system of the ordinary differential equation, in which the resulting coefficient matrix is complex symmetric and possesses the block Toeplitz structure. An exponential Runge–Kutta method is employed to solve such a system of the ordinary differential equation. Theoretically, the proposed method is second-order accuracy in space and fourth-order accuracy in time, respectively. In the practical implementation, the product of a block Toeplitz matrix exponential and a vector is calculated by the shift-invert Lanczos method. Meanwhile, the sectorial operator (the coefficient matrix) guarantees the fast approximation by the shift-invert Lanczos method. Numerical experiments are carried out to testify the theoretical results and demonstrate that the proposed method enjoys the excellent computational advantage.

Explicit pseudo two-step exponential Runge–Kutta methods for the numerical integration of first-order differential equations

This paper is devoted to the explicit pseudo two-step exponential Runge–Kutta (EPTSERK) methods for the numerical integration of first-order ordinary differential equations. These methods inherit the structure of explicit pseudo two-step Runge–Kutta methods and explicit exponential Runge–Kutta methods. We analyze the order conditions and the global errors of the new methods. The new methods are of order s + 1 with s-stages for some suitable nodes. Numerical experiments are reported to show the convergence and the efficiency of the new methods.

Efficient implementation of partitioned stiff exponential Runge-Kutta methods

Multiphysics systems are driven by multiple processes acting simultaneously, and their simulation leads to partitioned systems of differential equations. This paper studies the solution of partitioned systems of differential equations using exponential Runge-Kutta methods. We propose specific multiphysics implementations of exponential Runge-Kutta methods satisfying stiff order conditions that were developed in Hochbruck et al. (1998) [22] and Luan and Osterman (2014) [25]. We reformulate stiffly–accurate exponential Runge–Kutta methods in a way that naturally allows of the structure of multiphysics systems, and discuss their application to both component and additively partitioned systems. The resulting partitioned exponential methods only compute matrix functions of the Jacobians of individual components, rather than the Jacobian of the full, coupled system. We derive modified formulations of particular methods of order two, three and four, and apply them to solve a partitioned reaction-diffusion problem. The proposed methods retain full order for several partitionings of the discretized problem, including by components and by physical processes.

Asymptotical stability of numerical methods for semi-linear impulsive differential equations

This paper is concerned with asymptotical stability of a class of semi-linear impulsive ordinary differential equations. First of all, sufficient conditions for asymptotical stability of the exact solutions of semi-linear impulsive differential equations are provided. Under the sufficient conditions, some explicit exponential Runge–Kutta methods can preserve asymptotically stability without additional restriction on stepsizes. Moreover, it is proved that some explicit Runge–Kutta methods can preserve asymptotical stability without additional restriction on stepsizes under stronger conditions.

Numerical solution for multi-dimensional Riesz fractional nonlinear reaction–diffusion equation by exponential Runge–Kutta method

A spatial discretization of the Riesz fractional nonlinear reaction–diffusion equation by the fractional centered difference scheme leads to a system of ordinary differential equations, in which the resulting coefficient matrix possesses the symmetric block Toeplitz structure. An exponential Runge–Kutta method is employed to solve such a system of ordinary differential equations. In the practical implementation, the product of a block Toeplitz matrix exponential and a vector is calculated by the shift-invert Lanczos method. Meanwhile, the symmetric positive definiteness of the coefficient matrix guarantees the fast approximation by the shift-invert Lanczos method. Numerical results are given to demonstrate the efficiency of the proposed method.

Convergence of an Exponential Runge–Kutta Method for Non-smooth Initial Data

The paper presents error bounds for the second order exponential Runge-Kutta method for parabolic abstract linear time-dependent differential equations incorporating non-smooth initial data. As an example for this particular type of problems, the paper presents a spatial discretization of a partial integro-differential equation arising in financial mathematics, where non-smooth initial conditions occur in option pricing models. For this example, numerical studies of the convergence rate are given

Two-dimensional simulation of the damped Kuramoto–Sivashinsky equation via radial basis function-generated finite difference scheme combined with an exponential time discretization

We apply a numerical scheme based on a meshless method in space and an explicit exponential Runge-Kutta in time for the solution of the damped Kuramoto–Sivashinsky equation in two-dimensional spaces. The proposed meshless method is radial basis function-generated finite difference, which approximates the derivatives of the unknown function with respect to the spatial variables by a linear combination of the function values at given points in the domain and weights. Also, in this approach there is no need a mesh or triangulation for approximation. For each point, the weights are computed separately in its local sub-domain by solving a small radial basis function interpolant. Besides, a numerical algorithm based on singular value decomposition of the local radial basis function interpolation matrix [59] is applied to find the suitable shape parameter for each interpolation problem. We also consider an explicit time discretization based on exponential Runge–Kutta scheme such that its stability region is bigger than the classical form of Runge-Kutta method. Some numerical simulations are provided on the square, circular and annular domains to show the capability of the numerical scheme proposed here.

Half-Explicit Exponential Runge–Kutta Methods for Index-1 DAEs in Helicopter Simulation

In this paper we suggest a combination of exponential integrators and half-explicit Runge–Kutta methods for solving index-1 DAE systems with a stiff linear part in their differential equations. We discuss the behavior of the resulting half-explicit exponential Runge–Kutta (HEERK) methods for a simple numerical example and for a coupled rotor simulation. The coupled rotor simulation is based on a modular software design where all subsystems are modeled by ODEs in state-space form. By connecting the subsystems’ inputs and outputs we obtain an index-1 DAE system. Large terms in the system can be expressed as a stiff linear part which includes strong damping or oscillation terms as well as coefficients for the discretization of the rotor blades (3d beam equations). We show that the proposed HEERK methods can solve the resulting system efficiently with a reasonable timestep size.

Existence of solutions and numerical approximation of a non-local tumor growth model.

In order to model the evolution of a heterogeneous population of cancer stem cells and tumor cells, we analyse a nonlinear system of integro-differential equations. We provide an existence and uniqueness result by exploiting a suitable iterative scheme of functions which converge to the solution of the system. Then, we discretize the model and perform some numerical simulations. Numerical approximations are obtained by applying finite differences for space discretization and an exponential Runge-Kutta scheme for time integration. We exploit the numerical tool in order to investigate the effects that niches have on cancer development. In this respect, the numerical procedure is applied in the case when the function of cell redistribution is assumed to be spatially explicit. It allows for finding an approximate solution which is spatially inhomogeneous as time progresses. In this framework, numerical investigation may help in understanding the process of niche construction, which plays an important role in cancer population biology.

Exponential integrators with quadratic energy preservation for linear Poisson systems

Motivated by the recent paper (Wang and Wu, 2018 [12]), where the authors proposed a class of functionally-fitted energy-preserving integrators for Poisson systems, in this paper we aim at developing exponential integrators with quadratic energy preservation for the important class of linear Poisson systems. By rewriting the linear Poisson system in a new formulation, we apply the symplectic exponential Runge-Kutta methods and thus obtain a novel class of exponential integrators for the linear Poisson system. It is proved that the obtained exponential integrator can exactly preserve the quadratic energy. In comparison with the functionally-fitted energy-preserving methods whose numerical performance depends on the fitted frequency, the new exponential integrators are independent of the fitted frequency and convenient to use. Furthermore, we investigate the convergence of the fixed-point iteration used in the implementation of the proposed exponential integrator, and show that the convergence region of the stepsize is independent of the norm of the skew-symmetric matrix. Finally, numerical results with the Euler equations of a free rigid body soundly support the remarkable performance of the exponential integrator in both the global errors and the energy preservation.

A Second-Order Asymptotic-Preserving and Positivity-Preserving Exponential Runge--Kutta Method for a Class of Stiff Kinetic Equations

We introduce a second-order time discretization method for stiff kinetic equations. The method is asymptotic-preserving (AP) -- can capture the Euler limit without numerically resolving the small Knudsen number; and positivity-preserving -- can preserve the non-negativity of the solution which is a probability density function for arbitrary Knudsen numbers. The method is based on a new formulation of the exponential Runge-Kutta method and can be applied to a large class of stiff kinetic equations including the BGK equation (relaxation type), the Fokker-Planck equation (diffusion type), and even the full Boltzmann equation (nonlinear integral type). Furthermore, we show that when coupled with suitable spatial discretizations the fully discrete scheme satisfies an entropy-decay property. Various numerical results are provided to demonstrate the theoretical properties of the method.

D-convergence and conditional GDN-stability of exponential Runge–Kutta methods for semilinear delay differential equations

This paper is concerned with exponential Runge–Kutta methods with Lagrangian interpolation (ERKLMs) for semilinear delay differential equations (DDEs). Concepts of exponential algebraic stability and conditional GDN-stability are introduced. D-convergence and conditional GDN-stability of ERKLMs for semilinear DDEs are investigated. It is shown that exponentially algebraically stable and diagonally stable ERKLMs with stage order p, together with a Lagrangian interpolation of order q (q ≥ p), are D-convergent of order p. It is also shown that exponentially algebraically stable and diagonally stable ERKLMs are conditionally GDN-stable. Some examples of exponentially algebraically stable and diagonally stable ERKLMs of stage order one and two are given, and numerical experiments are presented to illustrate the theoretical results.