Partitioned Runge Kutta Research Articles

Projective integration methods in the Runge–Kutta framework and the extension to adaptivity in time

Projective Integration methods are explicit time integration schemes for stiff ODEs with large spectral gaps. In this paper, we show that all existing Projective Integration methods can be written as Runge–Kutta methods with an extended Butcher tableau including many stages. We prove consistency and order conditions of the Projective Integration methods using the Runge–Kutta framework. Spatially adaptive Projective Integration methods are included via partitioned Runge–Kutta methods. New time adaptive Projective Integration schemes are derived via embedded Runge–Kutta methods and step size variation while their accuracy, stability, convergence, and error estimators are investigated analytically and numerically.

An efficient method of finding new symplectic schemes for Hamiltonian mechanics problems with the aid of parametric Gröbner bases

The explicit symplectic difference schemes are considered for the numerical solution of molecular dynamics problems described by systems with separable Hamiltonians. A general method for finding symplectic schemes of high order of accuracy using parametric Gröbner bases, resultants, permutations of variables, and optimization techniques is proposed. The implementation of the method is described by the example of four-stage partitioned Runge–Kutta (PRK) schemes of the Forest–Ruth and Yoshida families. 97 new PRK schemes of Forest–Ruth family have been obtained. The error functionals for five new schemes FR47, FR50, FR51, FR52, and FR59 are smaller by more than two orders of magnitude than in the case of the fourth-order schemes, which were obtained previously in the explicit form in this family. In the family of Yoshida schemes, 115 new schemes have been found. Verification of the schemes was carried out on the Kepler's problem, which has the exact solution. Calculations of this task performed at large time steps showed that the FR50 scheme significantly exceeds in terms of accuracy the four-stage Forest–Ruth schemes known in the literature, the Verlet scheme, schemes from the Yoshida family Y7, Y95, Y110, and a six-stage fourth-order scheme.

A Splitting Nearly-Analytic Symplectic Partitioned Runge–Kutta Method for Solving 2D Elastic Wave Equations

In this paper, we present a new splitting nearly-analytic symplectic partitioned Runge–Kutta (SNSPRK) method for the two-dimensional (2D) elastic wave equations. It is an extension to elastic wave equation of our recent work on the locally one-dimensional nearly-analytic symplectic partitioned Runge–Kutta (LOD-NSPRK) method for the 2D acoustic wave equations. The method is based on the spatial differential operator-split technique, in which the resulting spatial discrete matrices are symmetric unlike the conventional nearly-analytic symplectic partitioned Runge–Kutta (NSPRK) method. The stability condition is given, which has more relax restriction for the time step than the NSPRK schemes. To show the performance of the new method, numerical experiments are given. Numerical results illustrate that the SNSPRK method has better long-term calculation capability as time proceeds than the NSPRK method.

Multirate partitioned Runge–Kutta methods for coupled Navier–Stokes equations

Earth system models are complex integrated models of atmosphere, ocean, sea ice, and land surface. Coupling the components can be a significant challenge due to the difference in physics, temporal, and spatial scales. This study explores multirate partitioned Runge–Kutta methods for the fluid–fluid interaction problem and demonstrates its parallel performance by using the PETSc library. We consider compressible Navier–Stokes equations with gravity coupled through a rigid-lid interface. Our large-scale numerical experiments reveal that multirate partitioned Runge–Kutta coupling schemes (1) can conserve total mass; (2) have second-order accuracy in time; and (3) provide favorable strong- and weak-scaling performance on modern computing architectures. We also show that the speedup factors of multirate partitioned Runge–Kutta methods match theoretical expectations over their base (single-rate) method.

A modified symplectic discontinuous Galerkin method for acoustic and elastic wave simulations

Numerically solving seismic wave equations is vital to large-scale forward modeling and full waveform inversion. In this paper, a new modified symplectic discontinuous Galerkin (MSDG) method is proposed to solve the acoustic and elastic equations. The MSDG method employs a symmetric interior penalty Galerkin formulation as the space discretization. The time discretization is based on a modified symplectic partitioned Runge–Kutta scheme with minimized phase error. Thus, the MSDG method has the advantages of high accuracy, being flexible to deal with complex geometric boundaries and internal structures, and stable for long time simulations. The numerical stability conditions, dispersion and dissipation are investigated in detail for the MSDG method. To validate the method, we carry out several numerical examples for solving the acoustic and elastic wave equations in various media. The numerical results show that the MSDG method can effectively suppress the numerical dispersion and is suitable for wavefield simulations.

A conservative fourth-order real space method for the (2+1)D Dirac equation

Modelling the time-dependent (2+1)D Dirac equation has recently gained importance since this equation effectively describes multiple condensed matter systems. To avoid the large dispersion errors of second-order real space schemes, a highly accurate method is presented here instead. The method utilises a fourth-order central difference on a staggered grid and an explicit symplectic Partitioned Runge–Kutta (PRK) time integrator. In contrast to traditional Runge–Kutta (RK) time stepping, no unphysical dissipation is introduced into the simulation. Moreover, it is demonstrated, both theoretically via Poisson maps and numerically, that the novel scheme has excellent conservation properties. Furthermore, the proposed numerical method is provably stable and the dispersion error is low and isotropic. Several interesting numerical examples are presented. Besides validating the advocated method, they also showcase its computational efficiency and low memory consumption.

A high-order linearly implicit energy-preserving Partitioned Runge-Kutta scheme for a class of nonlinear dispersive equations

<abstract><p>In this paper, we design a novel class of arbitrarily high-order, linearly implicit and energy-preserving numerical schemes for solving the nonlinear dispersive equations. Based on the idea of the energy quadratization technique, the original system is firstly rewritten as an equivalent system with a quadratization energy. The prediction-correction strategy, together with the Partitioned Runge-Kutta method, is then employed to discretize the reformulated system in time. The resulting semi-discrete system is high-order, linearly implicit and can preserve the quadratic energy of the reformulated system exactly. Finally, we take the Camassa-Holm equation as a benchmark to show the efficiency and accuracy of the proposed schemes.</p></abstract>

Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems

We propose efficient numerical methods for nonseparable non-canonical Hamiltonian systems which are explicit, K-symplectic in the extended phase space with long time energy conservation properties. They are based on extending the original phase space to several copies of the phase space and imposing a mechanical restraint on the copies of the phase space. Explicit K-symplectic methods are constructed for two non-canonical Hamiltonian systems. Numerical tests show that the proposed methods exhibit good numerical performance in preserving the phase orbit and the energy of the system over long time, whereas higher order Runge–Kutta methods do not preserve these properties. Numerical tests also show that the K-symplectic methods exhibit better efficiency than that of the same order implicit symplectic, explicit and implicit symplectic methods for the original nonseparable non-canonical systems. On the other hand, the fourth order K-symplectic method is more efficient than the fourth order Yoshida’s method, the optimized partitioned Runge–Kutta and Runge–Kutta–Nyström explicit K-symplectic methods for the extended phase space Hamiltonians, but less efficient than the the optimized partitioned Runge–Kutta and Runge–Kutta–Nyström extended phase space symplectic-like methods with the midpoint permutation.

High-order integrators for Lagrangian systems on homogeneous spaces via nonholonomic mechanics

In this paper, high-order numerical integrators on homogeneous spaces will be presented as an application of nonholonomic partitioned Runge–Kutta Munthe-Kaas (RKMK) methods on Lie groups. A homogeneous space M is a manifold where a group G acts transitively. Such a space can be understood as a quotient M≅G/H, where a closed Lie subgroup H, is the isotropy group of each point of M. The Lie algebra of G may be decomposed into g=m⊕h, where h is the subalgebra that generates H and m is a subspace spanning the tangent spaces of M. Thus, variational problems on M can be treated as nonholonomically constrained problems on G, by requiring variations to remain on m. Nonholonomic partitioned RKMK integrators are derived as a modification of those obtained by a discrete variational principle on Lie groups, and can be interpreted as obeying a discrete Chetaev principle. These integrators tend to preserve several properties of their purely variational counterparts.

Entropy–Preserving and Entropy–Stable Relaxation IMEX and Multirate Time–Stepping Methods

We propose entropy-preserving and entropy-stable partitioned Runge–Kutta (RK) methods. In particular, we extend the explicit relaxation Runge–Kutta methods to IMEX–RK methods and a class of explicit second-order multirate methods for stiff problems arising from scale-separable or grid-induced stiffness in a system. The proposed approaches not only mitigate system stiffness but also fully support entropy-preserving and entropy-stability properties at a discrete level. The key idea of the relaxation approach is to adjust the step completion with a relaxation parameter so that the time-adjusted solution satisfies the entropy condition at a discrete level. The relaxation parameter is computed by solving a scalar nonlinear equation at each timestep in general; however, as for a quadratic entropy function, we theoretically derive the explicit form of the relaxation parameter and numerically confirm that the relaxation parameter works the Burgers equation. Several numerical results for ordinary differential equations and the Burgers equation are presented to demonstrate the entropy-conserving/stable behavior of these methods. We also compare the relaxation approach and the incremental direction technique for the Burgers equation with and without a limiter in the presence of shocks.

An efficient symplectic stereo-modeling method for seismic inversion by using deep learning technique

AbstractThis paper proposes an efficient symplectic stereo-modeling (SSTEM) method for full waveform inversion (FWI) by using a deep learning technique. To solve the 2D acoustic equation, the SSTEM method uses a third-order optimal symplectic partitioned Runge–Kutta approach as a time-stepping method. An eighth-order stereo-modeling operator is used for spatial discretization. The SSTEM method is then expressed with a recurrent neural network (RNN). This is realized mainly because the time advancing format of the SSTEM method is similar to that of RNN, and they both use the information from the previous time step to obtain information from the current time step. With SSTEM as the forward modeling method, FWI is implemented using Tensorflow. The well-known adaptive moment estimation (Adam) optimizer and Nesterov adaptive moment estimation (Nadam) optimizer with mini-batch are used. The applicability of the developed code is also verified on GPUs. The numerical results show that the SSTEM method is more efficient and produces less numerical dispersion than the conventional finite-difference (FD) method when the same sampling rate in a wavelength is used. We compare several loss functions. The mean square (MSE) error and absolute (ABS) error loss functions are first tested. Another loss function that adds a physical differential operator to the original loss function is then considered. The FWI results show that this loss function has some improvements. Finally, we implement FWI on the complex Marmousi and SEG/EAGE models, and the inversion results demonstrate that the proposed method is suitable for seismic imaging in complex media.

Three kinds of novel multi-symplectic methods for stochastic Hamiltonian partial differential equations

Stochastic Hamiltonian partial differential equations, which possess the multi-symplectic conservation law, are an important and fairly large class of systems. The multi-symplectic methods inheriting the geometric features of stochastic Hamiltonian partial differential equations provide numerical approximations with better numerical stability, and are of vital significance for obtaining correct numerical results. In this paper, we propose three novel multi-symplectic methods for stochastic Hamiltonian partial differential equations based on the local radial basis function collocation method, the splitting technique and the partitioned Runge–Kutta method. Specific numerical methods are presented for nonlinear stochastic wave equations, stochastic nonlinear Schrödinger equations, stochastic Korteweg-de Vries equations and stochastic Maxwell equations. We take stochastic wave equations as examples to perform numerical experiments, which indicate the validity of the proposed methods.

Second-order symplectic PRK scheme for ground-penetrating radar simulations in dispersive materials

An accurate and effective forward model is helpful in interpreting measured ground-penetrating radar (GPR) data. In this paper, an explicit second-order symplectic partitioned Runge–Kutta scheme (SPRK) with Higdon's absorbing boundary condition is developed to simulate GPR wave propagation in dispersive media. For this purpose, the first-order Debye dispersion model is successfully applied to the second-order SPRK method. A single-layer dispersive medium model and a three-layer pavement structure model are simulated, in order to verify the accuracy and efficiency of the second-order SPRK method, as well as to clarify the effect of dispersion on the GPR echo signal. The numerical simulation results demonstrate that the proposed algorithm not only can maintain the simulation accuracy, but can also save about 19% of the calculation time, when compared with the FDTD algorithm, in a dispersive medium. We also obtain the GPR profile of a pavement structure with a circular cavity disease and an underground pipe structure in a dispersive medium, and the simulation results show that the dispersion characteristics of the materials lead to high absorption of the GPR electromagnetic waves. Finally, field experiment data are used to further verify the applicability of the algorithm. The numerical simulations and experiments indicate that the dispersion of the medium must be considered in GPR numerical simulation, and the authenticity of the simulated image must be ensured.

A Note on the Construction of Explicit Symplectic Integrators for Schwarzschild Spacetimes

Abstract In recent publications, the construction of explicit symplectic integrators for Schwarzschild- and Kerr-type spacetimes is based on splitting and composition methods for numerical integrations of Hamiltonians or time-transformed Hamiltonians associated with these spacetimes. Such splittings are not unique but have various options. A Hamiltonian describing the motion of charged particles around the Schwarzschild black hole with an external magnetic field can be separated into three, four, and five explicitly integrable parts. It is shown through numerical tests of regular and chaotic orbits that the three-part splitting method is the best of the three Hamiltonian splitting methods in accuracy. In the three-part splitting, optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators exhibit the best accuracies. In fact, they are several orders of magnitude better than the fourth-order Yoshida algorithms for appropriate time steps. The first two algorithms have a small additional computational cost compared with the latter ones. Optimized sixth-order partitioned Runge–Kutta and Runge–Kutta–Nyström explicit symplectic integrators have no dramatic advantages over the optimized fourth-order ones in accuracy during long-term integrations due to roundoff errors. The idea of finding the integrators with the best performance is also suitable for Hamiltonians or time-transformed Hamiltonians of other curved spacetimes including Kerr-type spacetimes. When the numbers of explicitly integrable splitting sub-Hamiltonians are as small as possible, such splitting Hamiltonian methods would bring better accuracies. In this case, the optimized fourth-order partitioned Runge–Kutta and Runge–Kutta–Nyström methods are worth recommending.

Symplectic P-stable additive Runge—Kutta methods

<p style='text-indent:20px;'>Classical symplectic partitioned Runge–Kutta methods can be obtained from a variational formulation where all the terms in the discrete Lagrangian are treated with the same quadrature formula. We construct a family of symplectic methods allowing the use of different quadrature formulas (primary and secondary) for different terms of the Lagrangian. In particular, we study a family of methods using Lobatto quadrature (with corresponding Lobatto IIIA-B symplectic pair) as a primary method and Gauss–Legendre quadrature as a secondary method. The methods have the same implicitness as the underlying Lobatto IIIA-B pair, and, in addition, they are <i>P-stable</i>, therefore suitable for application to highly oscillatory problems.</p>

Higher-order symplectic integration techniques for molecular dynamics problems

The explicit symplectic difference schemes with a number of stages from 1 to 5 are considered for the numerical solution of molecular dynamics problems described by systems with separable Hamiltonians. A general method for finding symplectic schemes of high order of accuracy using Gröbner bases is proposed. It is shown that it is possible to significantly increase the accuracy of symplectic schemes without increasing the number of stages by obtaining all possible real schemes with the aid of Gröbner bases. For the numbers of stages 2, 3, and 4, the parameters of Runge–Kutta–Nyström (RKN) schemes and partitioned Runge–Kutta (PRK) schemes are obtained using the Gröbner basis technique. It is shown that there exists only one explicit invertible (symmetric) two-stage RKN scheme. In the cases of the vanishing Vandermonde determinant, 20 new real four-stage RKN schemes and 18 new PRK schemes of Forest–Ruth and Yoshida were found in the analytic form with the aid of Gröbner bases. In the cases of the nonzero Vandermonde determinant, four symmetric four-stage RKN schemes were derived in the analytic form with the aid of the Gröbner basis technique. In addition, for the number of stages 4 and 5, new nonsymmetric schemes were found using the Nelder–Mead numerical optimization method. In particular, four new schemes were obtained for the number of stages 4. For the number of stages 5, three new schemes were obtained in addition to four schemes known earlier. For each specific number of stages, a scheme has been found that is the best in terms of the minimum of the leading term of the approximation error. The relations have been established between the polynomials entering the expansions of the errors of each scheme for the momentum and for the particle coordinate. Verification of the schemes was carried out on a problem that has an exact solution. It is shown that the symplectic five-stage RKN scheme provides a more accurate conservation of the total energy balance of the particle system than schemes of lower orders of accuracy. The stability studies of the schemes were performed using the Mathematica software package.

Implicit Multirate GARK Methods

This work considers multirate generalized-structure additively partitioned Runge–Kutta methods for solving stiff systems of ordinary differential equations with multiple time scales. These methods treat different partitions of the system with different timesteps for a more targeted and efficient solution compared to monolithic single rate approaches. With implicit methods used across all partitions, methods must find a balance between stability and the cost of solving nonlinear equations for the stages. In order to characterize this important trade-off, we explore multirate coupling strategies, problems for assessing linear stability, and techniques to efficiently implement Newton iterations for stage equations. Unlike much of the existing multirate stability analysis which is limited in scope to particular methods, we present general statements on stability and describe fundamental limitations for certain types of multirate schemes. New implicit multirate methods up to fourth order are derived, and their accuracy and efficiency properties are verified with numerical tests.

Generalization of partitioned Runge–Kutta methods for adjoint systems

This study computes the gradient of a function of numerical solutions of ordinary differential equations (ODEs) with respect to the initial condition. The adjoint method computes the gradient approximately by solving the corresponding adjoint system numerically. In this context, Sanz-Serna [SIAM Rev., 58 (2016), pp. 3–33] showed that when the initial value problem is solved by a Runge–Kutta (RK) method, the gradient can be exactly computed by applying an appropriate RK method to the adjoint system. Focusing on the case where the initial value problem is solved by a partitioned RK (PRK) method, this paper presents a numerical method, which can be seen as a generalization of PRK methods, for the adjoint system that gives the exact gradient.

A general symplectic scheme with three free parameters and its applications

Symplectic schemes applied to Hamiltonian systems have prominent advantages for the preservation of qualitative properties of the flow. Three types of symplectic methods, which contain the symplectic Euler, implicit midpoint and Störmer–Verlet methods, are simplest and widely used in actual calculations. In this paper, we introduce a simple symplectic scheme with three free parameters, which covers these three methods and has the same behaviors as them. The symplecticity of this scheme is verified from partitioned Runge–Kutta methods and variational integrators. In addition, we get a second-order symplectic scheme with two free parameters and a symmetric symplectic scheme with a free parameter. By adjusting the parameter at each time step, we get a second-order energy and quadratic invariants preserving method. The effectiveness of all schemes is demonstrated by numerical tests.

Learning Adaptive Regularization for Image Labeling Using Geometric Assignment

We study the inverse problem of model parameter learning for pixelwise image labeling, using the linear assignment flow and training data with ground truth. This is accomplished by a Riemannian gradient flow on the manifold of parameters that determines the regularization properties of the assignment flow. Using the symplectic partitioned Runge–Kutta method for numerical integration, it is shown that deriving the sensitivity conditions of the parameter learning problem and its discretization commute. A convenient property of our approach is that learning is based on exact inference. Carefully designed experiments demonstrate the performance of our approach, the expressiveness of the mathematical model as well as its limitations, from the viewpoint of statistical learning and optimal control.