Exponential Integration Methods Research Articles

Meshfree generalized multiscale exponential integration method for parabolic problems

Meshfree generalized multiscale exponential integration method for parabolic problems

Improved uniform error bounds of a Lawson-type exponential wave integrator method for the Klein-Gordon-Dirac equation

For the Klein-Gordon-Dirac equation (KGDE) with small coupling constant ε∈(0,1], we propose a Lawson-type exponential wave integrator Fourier pseudo-spectral (LEWIFP) method and establish the improved uniform error bounds in the time domain at O(1/ε). We first convert the KGDE to a coupled system and then consider LEWIFP method for the coupled system. The LEWIFP method is proved to be time symmetric which is an important structure in numerical geometric integration. Through careful and rigorous convergence analysis, we establish the error bounds O(hm+ετ2+τ0m) for the full-discretization, where m is determined by the regularity conditions. If the solution is sufficiently smooth with m sufficiently large, we obtain the errors with improved uniform bounds at O(hm+ετ2) in the long-time domain up to O(1/ε). These error bounds are much better than the classical bounds O(hm+τ2) provided by the traditional analysis for the non-Lawson-type exponential wave integrators equipped with Fourier pseudo-spectral method. Combined with the classical analysis tools such as mathematical induction and energy method, we complete our error analysis by adopting the regularity compensation oscillation (RCO) technique which controls the high frequency modes by the regularity of the solution and low frequency modes by phase cancellation. By applying the LEWIFP method to some problems, we show the numerical results to support our error bounds. In addition, the numerical results also show that the discrete mass and energy are stable in the time domain which is long enough. Finally we extend our method to the oscillatory problem.

Variable Time-stepping Exponential Integrators for Chemical Reactors with Analytical Jacobians

Chemical combustion problems are known to be stiff and therefore difficult to efficiently integrate in time when numerically simulated. Implicit methods, such as backwards differentiation formula (BDF), are widely considered to be the state-of-the-art methods owing their capability of taking relatively large time-steps while maintaining accurate combustion characteristics. Exponential time integration methods have recently demonstrated the ability to accurately and efficiently solve large scale systems of ordinary differential equations. This study introduces a novel adaptive time stepping exponential integrator named EPI3V. Its performance is measured on spatially homogeneous isobaric reactive mixtures involving three hydrocarbon fuel mechanisms. The full combustion process is simulated using gas compositions with sufficient temperature to obtain auto-ignition. Simulations are run until the steady state is obtained, then a comparison of the computational efficiency and accuracy between a BDF and EPI3V method is made. The novel EPI3V method exhibits comparable computational efficiency to a well-established implementation of the variable time-stepping BDF implicit methods for two of the mechanisms investigated. In certain situations it even demonstrates a slight advantage over the implicit solver. However, in one specific case, the EPI3V shows relative performance degradation compared to the implicit method, but it still converges for this case. These results indicate that exponential time integration methods may be applicable to a larger variety of combustion problems.

Improved uniform error bounds on an exponential wave integrator method for the nonlinear Schrödinger equation with wave operator and weak nonlinearity

In this paper, we establish the improved uniform error bounds for the Lawson-type exponential wave integrator Fourier pseudo-spectral (LEWIFP) method for the nonlinear Schrödinger equation with wave operator (NLSW) and weak nonlinearity characterized by a small constant ε∈(0,1]. We first convert the NLSW to a coupled system and then consider the LEWIFP method for the coupled system. The LEWIFP method is proved to be time symmetric and mass-conservative which is very important from the point of view of numerical geometric integration. Through careful and rigorous convergence analysis, we establish improved uniform error bounds for the full-discretization at O(hm+ε2pτ2) in the long-time domain up to O(1/ε2p) where m is determined by the regularity conditions. These error bounds are much better than the classical error bounds O(hm+τ2) provided by the traditional error analysis. In error analysis, in addition to the classical tools such as mathematical induction and energy method, we adopt the regularity compensation oscillation (RCO) technique to analyze the accumulation of errors carefully. The numerical experiments support our error estimates and demonstrate the discrete mass-conservation. In addition, the numerical results show that the discrete energy is stable in the time domain up to O(1/ε2p).

Spectrally-tuned compact finite-difference schemes with domain decomposition and applications to numerical relativity

Compact finite-difference (FD) schemes specify derivative approximations implicitly, thus to achieve parallelism with domain-decomposition suitable partitioning of linear systems is required. Consistent order of accuracy, dispersion, and dissipation is crucial to maintain in wave propagation problems such that deformation of the associated spectra of the discretized problems is not too severe. In this work we consider numerically tuning spectral error, at fixed formal order of accuracy to automatically devise new compact FD schemes. Grid convergence tests indicate error reduction of at least an order of magnitude over standard FD. A proposed hybrid matching-communication strategy maintains the aforementioned properties under domain-decomposition. Under evolution of linear wave-propagation problems utilizing exponential integration or explicit Runge-Kutta methods improvement is found to remain robust. A first demonstration that compact FD methods may be applied to the Z4c formulation of numerical relativity is provided where we couple our header-only, templated C++ implementation to the highly performant GR-Athena++ code. Evolving Z4c on test-bed problems shows at least an order in magnitude reduction in phase error compared to FD for propagated metric components. Stable binary-black-hole evolution utilizing compact FD together with improved convergence is also demonstrated.

Improved uniform error bounds of Lawson-type exponential integrator method for long-time dynamics of the high-dimensional space fractional sine-Gordon equation

Improved uniform error bounds of Lawson-type exponential integrator method for long-time dynamics of the high-dimensional space fractional sine-Gordon equation

Improved uniform error bounds of an exponential wave integrator method for the Klein–Gordon–Schrödinger equation with the small coupling constant

Improved uniform error bounds of an exponential wave integrator method for the Klein–Gordon–Schrödinger equation with the small coupling constant

Structure‐preserving exponential wave integrator methods and the long‐time convergence analysis for the Klein‐Gordon‐Dirac equation with the small coupling constant

AbstractRecently, the numerical methods for long‐time dynamics of PDEs with weak nonlinearity (or small potentials) have received more and more attention. For the Klein‐Gordon‐Dirac (KGD) equation with the small coupling constant , we propose two time symmetric and structure‐preserving exponential wave integrator Fourier pseudo‐spectral (TSSPEWIFP) methods under periodic boundary conditions. The new methods are proved to preserve the energy in discrete level and in addition the first one preserves the modified discrete mass. Through rigorous error analysis, we establish uniform error bounds of the numerical solutions at up to the time at for where and are the mesh size and time step, respectively, and depends on the regularity conditions. Compared with the results of existing numerical analysis, our analysis has the advantage of showing the long time numerical errors for the KGD equation with the small coupling constant. The tools for error analysis mainly include cut‐off technique and the standard energy method. We also extend the results on error bounds, structure‐preservation and time symmetry to the oscillatory KGD equation with wavelength at in time which is equivalent to the KGD equation with small coupling constant. Numerical experiments confirm that the theoretical results are correct. Our methods are novel because that to the best of our knowledge there has not been any energy‐preserving exponential wave integrator method and any relevant long time analysis for the KGD equation.

Hybrid uncertainty propagation and reliability analysis using direct probability integral method and exponential convex model

Uncertainty propagation and reliability evaluation, being the crucial parts of engineering system analysis, play vital roles in safety assessment. How to reasonably consider the complex multisource uncertainty behavior in both static and dynamic systems is paramount to ensuring their safe operation. However, there is a significant lack of research on aleatory and epistemic uncertainties for both static and dynamic systems. To this end, a new hybrid exponential model is proposed by combining probabilistic and non-probabilistic exponential models, which aims to accurately measure the uncertainty propagation and reliability evaluation problem with aleatory and epistemic uncertainties for static and dynamic systems. The proposed hybrid exponential model consists of nested double optimization loops. The outer loop performs a probabilistic analysis based on the direct probability integral method, and the inner loop performs a non-probabilistic computation. Then, a new hybrid exponential probability integral method is developed to effectively perform uncertainty propagation and reliability analysis. Finally, four examples, including two static and two dynamic examples with complex performance functions, are tested. The results indicate that the proposed hybrid exponential model offers a universal tool for uncertainty quantification in static and dynamic systems. Moreover, the hybrid exponential probability integral method can accurately and efficiently obtain the upper and lower bounds of the probability density function and cumulative distribution function.

Along-the-path exponential integration for Floquet stability analysis of wind turbines

Traditionally, stability assessment of wind turbines has been performed by eigenanalysis of the azimuthally-averaged linearized system after applying the Multi-Blade Coordinate (MBC) transformation. However, due to internal or external anisotropy, the MBC transform does not produce an exact Linear Time-Invariant (LTI) system, and a Floquet analysis is required to capture the influence of all periodic terms, leading to a more accurate stability analysis. In this paper exponential integration methods that use system linearizations at different blade azimuth positions are used to integrate the perturbed system state and compute the Floquet monodromy matrix. The proposed procedure is assessed for a simple 6 DOF wind turbine model and a more complex aeroelastic model of a 5MW onshore wind turbine. The defined along-the-path or moving-point exponential integrator is found to be the suitable in order to perform a Floquet stability analysis even using a coarse linearization grid.

Mode-displacement method for structural dynamic analysis of bio-inspired structures: A palm-tree stem subject to wind effects

ABSTRACT Biological materials (orthotropic materials), like wood, can offer good mechanical properties with a minimum amount of material, making their internal structure the suitable one to be applied on bio-inspired structures. The knowledge of the exceptional structural performance of palm trees, and specially its response to different loading conditions, provides useful information when lightweight structures with high slenderness ratio are desired. Recent researches focused on the analysis of palm trees subject to static loading conditions, ignoring the fluctuating nature of the wind speed. The purpose of this study is to simulate in a computational efficient way the effect of dynamic loading conditions applied on palm trees. Using the mode displacement method, the number of degrees of freedom of a dynamic finite element analysis can be drastically reduced with a minimal loss of accuracy. It was applied to simulate the behavior of structures comprised of an orthotropic material subject to a stochastic dynamic load. The influence of the number of selected degrees of freedom has also been studied. In addition, an exponential integration method is proposed to perform the time integration procedure. The results obtained show that a properly reduced model suitably represents the full finite element model without any appreciable loss of accuracy; it is also shown that computational cost can be drastically reduced. This method could give an appropriate computational representation of the behavior of orthotropic structures, and it could be used for studying more complex bio-inspired structures.

Exponential integrator method for solving the nonlinear Helmholtz equation

<abstract><p>In this paper, we study the exponential integrator method (EIM) for solving the nonlinear Helmholtz equation (NLHE). As the wave number or the characteristic coefficient in the nonlinear term is large, the NLHE becomes a highly oscillatory and indefinite nonlinear problem, which makes most of numerical methods lose their expected computational effects. Based on the shooting method, the NLHE is firstly transformed into an initial-value-type problem. Then, the EIM is utilized for solving the deduced problem, by which we not only can capture the oscillation very well, but also avoid to search the nonlinear iteration method and to solve indefinite linear equations at each iteration step. Therefore, the high accuracy simulations with relative large physical parameters in the NLHE become possible and lots of computational costs can be saved. Some numerical examples, including the extension to the nonlinear Helmholtz system, are shown to verify the accuracy and efficiency of the proposed method.</p></abstract>

Deflated Restarting of Exponential Integrator Method with an Implicit Regularization for Efficient Transient Circuit Simulation

Exponential integrator (EI) method based on Krylov subspace approximation is a promising method for large-scale transient circuit simulation. However, it suffers from the singularity problem and consumes large subspace dimensions for stiff circuits when using the ordinary Krylov subspace. Restarting schemes are commonly applied to reduce the subspace dimension, but they also slow down the convergence and degrade the overall computational efficiency. In this paper, we first devise an implicit and sparsity-preserving regularization technique to tackle the singularity problem facing EI in the ordinary Krylov subspace framework. Next, we analyze the root cause of the slow convergence of the ordinary Krylov subspace methods when applied to stiff circuits. Based on the analysis, we propose a deflated restarting scheme, compatible with the above regularization technique, to accelerate the convergence of restarted Krylov subspace approximation for EI methods. Numerical experiments demonstrate the effectiveness of the proposed regularization technique, and up to 50% convergence improvements for Krylov subspace approximation compared to the non-deflated version.

A New Algorithm with Square Conservative Exponential Integral for Transport Equation

AbstractThe square conservative exponential integral method (SCEIM) is proposed for transport problems on the sphere. The method is a combination of the square conservation algorithm and the exponential integral method. The main emphasis in the development of SCEIM is on conservation, positive-definite, and reversibility as well as achieving comparable accuracy to other published schemes. The most significant advantage of SCEIM is to change the forward model to the backward model by setting a negative time step, and the backward model can be used to solve the inverse problem. Moreover, the polar problem is significantly improved by using a simple effective central skip-point difference scheme without major penalty on the overall effectiveness of SCEIM. To demonstrate the effectiveness and generality of the SCEIM, this method is evaluated by standard cosine bell tests and deformational flow tests. The numerical results show that SCEIM is a time-convergence method as well as a grid-convergence method, and has a strong shape-preserving ability. In the tests of the inverse problem, the sharp fronts are successfully regressed back into their initial weak fronts and the cosine bells move against the wind direction and return to the initial position with high accuracy. The numerical results of forward simulations are compared with those of published schemes, the total mass conservation, and error norms are competitive in term of accuracy.

A New Precursor Integral Method for Solving Space-Dependent Kinetic Equations in Neutronic and Thermal-Hydraulic Coupling System

The accurate prediction of the neutronic and thermal-hydraulic coupling system transient behavior is important in nuclear reactor safety analysis, where a large-scale nonlinear coupling system with strong stiffness should be solved efficiently. In order to reduce the stiffness and huge computational cost in the coupling system, the high-performance numerical techniques for solving delayed neutron precursor equation are a key issue. In this work, a new precursor integral method with an exponential approximation is proposed and compared with widely used Taylor approximation-based precursor integral methods. The truncation errors of exponential approximation and Taylor approximation are analyzed and compared. Moreover, a time control technique is put forward which is based on flux exponential approximation. The procedure is tested in a 2D neutron kinetic benchmark and a simplified high-temperature gas-cooled reactor-pebble bed module (HTR-PM) multiphysics problem utilizing the efficient Jacobian-free Newton–Krylov method. Results show that selecting appropriate flux approximation in the precursor integral method can improve the efficiency and precision compared with the traditional method. The computation time is reduced to one-ninth in the HTR-PM model under the same accuracy when applying the exponential integral method with the time adaptive technique.

A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations

<p style='text-indent:20px;'>In this paper, we propose a conservative semi-Lagrangian finite difference (SLFD) weighted essentially non-oscillatory (WENO) scheme, based on Runge-Kutta exponential integrator (RKEI) method, to solve one-dimensional scalar nonlinear hyperbolic equations. Conservative semi-Lagrangian schemes, under the finite difference framework, usually are designed only for linear or quasilinear conservative hyperbolic equations. Here we combine a conservative SLFD scheme developed in [<xref ref-type="bibr" rid="b21">21</xref>], with a high order RKEI method [<xref ref-type="bibr" rid="b7">7</xref>], to design conservative SLFD schemes, which can be applied to nonlinear hyperbolic equations. Our new approach will enjoy several good properties as the scheme for the linear or quasilinear case, such as, conservation, high order and large time steps. The new ingredient is that it can be applied to nonlinear hyperbolic equations, e.g., the Burgers' equation. Numerical tests will be performed to illustrate the effectiveness of our proposed schemes.</p>

Transparent boundary conditions and numerical computation for singularly perturbed telegraph equation on unbounded domain

In this paper, we study the numerical solution for the singularly perturbed telegraph equation (SPTE) on unbounded domain. Firstly, we investigate the first consistent effective asymptotic expansion for the solution of SPTE by the asymptotic analysis and obtain that the solutions of SPTE have an initial layer near $$t=0$$. Next, we introduce the artificial boundaries $$\varGamma _{\pm }$$ to get a finite computational domain $$\varOmega _0$$ and derive the transparent boundary conditions on $$\varGamma _{\pm }$$ for SPTE. Hence, we can reduce the original problem to an initial-boundary value problem (IBVP) on the bounded domain $$\varOmega _0$$, and then the equivalence between the original problem and the IBVP on $$\varOmega _0$$ is proved. In addition, we propose a Crank–Nicolson Galerkin scheme to solve the reduced problem. Furthermore, we use the exponential wave integrator method to deal with the initial layer. We also analyze the stability and convergence of the Crank–Nicolson Galerkin scheme. Finally, some numerical examples validate our theoretical results and show the efficiency and reliability of the transparent boundary conditions and the Crank–Nicolson Galerkin scheme.

Two-dimensional wave equation solved by generalized alternating flux based local discontinuous Galerkin method

The wave propagation is often carried out in complex geological structures. Solving the wave propagation problem effectively in inhomogeneous medium is of great interest and has many applications in physics and engineering. In this paper, the local discontinuous Galekin (LDG) method is applied to the numerical solution of the second-order wave equation. Firstly, the auxiliary variables are introduced, and the second-order wave equations are written as a system of first-order partial differential equations. Then the discontinuous Galerkin format is applied to the corresponding linearized wave equations and adjoint equations. We consider the triangulation in this paper. In order to ensure that the discrete format satisfies the energy conservation, the generalized alternating flux is chosen on the element boundary. We proves that the LDG method satisfies the energy conservation. The exponential integral factor method is used in time discretization. In order to improve the computational efficiency, the Krylov subspace method is used to approximate the product of the exponential matrix and the vector. Numerical examples with exact solutions are given in numerical experiments. The numerical results verify the numerical precision and energy conservation of the LDG method. In addition, the calculation of inhomogeneous medium and complex computational regions are considered. The results show that the LDG method is suitable for simulation of complex structures and propagation in multi-scale structured medium.

Exponential integrator methods for systems of non-linear space-fractional models with super-diffusion processes in pattern formation

Nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in Science and Engineering. However, solving these type of models is computationally demanding. In this paper, we propose an exponential integrator method for space fractional models as an attractive and easy-to-code alternative for other existing second-order exponential integrator methods. This scheme is based on using a real distinct poles discretization for the underlying matrix exponentials. One of the major benefits of the proposed scheme is that the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency. The scheme is established to be second-order convergent; and proven to be robust for nonlinear space fractional reaction–diffusion problems involving non-smooth initial data. Our approach is exhibited by solving a system of two-dimensional problems which exhibits pattern formation and has applications in cell-division. Empirically, super-diffusion processes are displayed by investigating the effect of the fractional power of the underlying Laplacian operator on the pattern formation found in these models. Furthermore, the superiority of our method over competing second order ETD schemes, BDF2 scheme, and IMEX schemes is demonstrated.

An Exponential Time Integrator for the Incompressible Navier--Stokes Equation

We present an exponential time integration method for the incompressible Navier--Stokes equation. An essential step in our procedure is the treatment of the pressure by applying a divergence-free projection to the momentum equation. The differential-algebraic equation for the discrete velocity and pressure is then reduced to a conventional ordinary differential equation that can be solved with the proposed exponential integrator. A promising feature of exponential time integration is its potential time parallelism within the Paraexp algorithm. We demonstrate that our approach leads to parallel speedup assuming negligible parallel communication.