Partitioned Runge-Kutta Methods 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.

Symplectic time-domain finite element method (STD-FEM) for room acoustic modeling

A new, extended symplectic time-domain finite element method (STD-FEM) is proposed for room acoustic modeling. As a mathematical model for sound propagation and reflection, the classical time-domain wave equations are used, which can be extended with air absorption. Frequency-dependent locally reactive boundary conditions are also introduced to the model. Spatially, a third-order tensor product type spectral element discretization is applied, which allows us to use explicit time steps. For this purpose, a partitioned Runge-Kutta method is employed, which is an extension of a third-order symplectic time-discretization. The method is validated on a number of examples and the performance and stability benefits are presented. We present the implementation of the method in the soundy.ai application.

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 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>

Optimal Control of Closed Quantum Systems via B-Splines with Carrier Waves

We consider the optimal control problem of determining electromagnetic pulses for implementing logical gates in a closed quantum system, where the Hamiltonian models the dynamics of coupled superconducting qudits. The quantum state is governed by Schrodinger's equation, which we formulate in terms of the real and imaginary parts of the state vector and solve by the Stormer-Verlet scheme, which is a symplectic partitioned Runge-Kutta method. A novel parameterization of the control functions based on B-splines with carrier waves is introduced. The carrier waves are used to trigger the resonant frequencies in the system Hamiltonian, and the B-spline functions specify their amplitude and phase. This approach allows the number of control parameters to be independent of, and significantly smaller than, the number of time steps for integrating Schrodinger's equation. We present numerical examples of how the proposed technique can be combined with an interior point L-BFGS algorithm for realizing quantum gates, and generalize our approach to calculate risk-neutral controls that are resilient to noise in the Hamiltonian model. The proposed method is also shown to compare favorably with QuTiP/pulse\_optim and Grape-Tensorflow.

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.

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. Concrete numerical methods are presented for nonlinear stochastic wave equations, stochastic nonlinear Schr\"odinger 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.

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>

Multisymplectic Hamiltonian Variational Integrators

Variational integrators have traditionally been constructed from the perspective of Lagrangian mechanics, but there have been recent efforts to adopt discrete variational approaches to the symplectic discretization of Hamiltonian mechanics using Hamiltonian variational integrators. In this paper, we will extend these results to the setting of Hamiltonian multisymplectic field theories. We demonstrate that one can use the notion of Type II generating functionals for Hamiltonian partial differential equations as the basis for systematically constructing Galerkin Hamiltonian variational integrators that automatically satisfy a discrete multisymplectic conservation law, and establish a discrete Noether's theorem for discretizations that are invariant under a Lie group action on the discrete dual jet bundle. In addition, we demonstrate that for spacetime tensor product discretizations, one can recover the multisymplectic integrators of Bridges and Reich, and show that a variational multisymplectic discretization of a Hamiltonian multisymplectic field theory using spacetime tensor product Runge–Kutta discretizations is well-defined if and only if the partitioned Runge-Kutta methods are symplectic in space and time.

Variational integrator graph networks for learning energy-conserving dynamical systems.

Recent advances show that neural networks embedded with physics-informed priors significantly outperform vanilla neural networks in learning and predicting the long-term dynamics of complex physical systems from noisy data. Despite this success, there has only been a limited study on how to optimally combine physics priors to improve predictive performance. To tackle this problem we unpack and generalize recent innovations into individual inductive bias segments. As such, we are able to systematically investigate all possible combinations of inductive biases of which existing methods are a natural subset. Using this framework we introduce variational integrator graph networks-a novel method that unifies the strengths of existing approaches by combining an energy constraint, high-order symplectic variational integrators, and graph neural networks. We demonstrate, across an extensive ablation, that the proposed unifying framework outperforms existing methods, for data-efficient learning and in predictive accuracy, across both single- and many-body problems studied in the recent literature. We empirically show that the improvements arise because high-order variational integrators combined with a potential energy constraint induce coupled learning of generalized position and momentum updates which can be formalized via the partitioned Runge-Kutta method.

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.

Numerical investigation of stochastic canonical Hamiltonian systems by high order stochastic partitioned Runge-Kutta methods

In this paper, a family of arbitrary high order quadratic invariants and energy conservation parametric stochastic partitioned Runge-Kutta methods (SPRK) are constructed for stochastic canonical Hamiltonian systems where the parameters depend on some truncated random variables, step size and numerical solutions. We first apply the P-series and bi-coloured trees theory to analyze the mean-square and weak convergence order conditions of SPRK methods solving a class of single integrand stochastic differential equations. Then, a class of SPRK methods with parameters are obtained by means of W-transform and technique of truncated Wiener increments, and we prove that the methods are symplectic. Combining with order conditions, there exists a special parameter α* which enables convergence order in each iteration and can preserve the energy of the stochastic canonical Hamiltonian systems. Finally, the representative stochastic canonical Hamiltonian systems are selected to verify the good performance of the proposed parameter methods.

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.

A third order, implicit, finite volume, adaptive Runge–Kutta WENO scheme for advection–diffusion equations

A finite volume approximation of the scalar hyperbolic conservation law or advection–diffusion equation is given. In the context of the method of lines, the space discretization uses weighted essentially non oscillatory (WENO) reconstructions with adaptive order (WENO-AO), and the time evolution uses implicit Runge–Kutta methods. Therefore the timestep may be larger than the CFL timestep. To reduce oscillation in the solution, ideas related to spatially partitioned Runge–Kutta methods are used. An adaptive Runge–Kutta method is developed that blends the L-stable, third order, implicit Radau IIA method with the composite backward Euler method using a weighting procedure inspired from spatial WENO methods. The weighting procedure requires a smoothness indicator, and several possibilities are considered, although one is perhaps seen to be preferred. The overall scheme is proven to maintain third order accuracy when the solution is smooth. When the solution has a discontinuity, the scheme is shown computationally to be third order accurate away from shocks, and to achieve the overall accuracy of the backward Euler method. Numerical examples show that the adaptive Runge–Kutta method reduces oscillations in the solution. Moreover, the resulting scheme is shown to be unconditionally L-stable for smooth solutions to the linear problem.

Symplecticness conditions of some low order partitioned methods for non-autonomous Hamiltonian systems

We consider the application of partitioned Runge-Kutta (PRK) methods to non-autonomous Hamiltonian systems. Necessary and sufficient conditions for the symplecticness of PRK methods are given, more particularly for two low order PRK methods: the partitioned (explicit-implicit) Euler method and the 2-stage Lobatto IIIA-B PRK method. Both methods are often the basis of composition schemes of higher order. In particular for irreducible PRK methods we show the necessity for the nodes of the two underlying PRK methods to be equal.

A nearly analytic symplectic partitioned Runge–Kutta method based on a locally one‐dimensional technique for solving two‐dimensional acoustic wave equations

ABSTRACTIn this paper, we develop a new nearly analytic symplectic partitioned Runge–Kutta method based on locally one‐dimensional technique for numerically solving two‐dimensional acoustic wave equations. We first split two‐dimensional acoustic wave equation into the local one‐dimensional equations and transform each of the split equations into a Hamiltonian system. Then, we use both a nearly analytic discrete operator and a central difference operator to approximate the high‐order spatial differential operators, which implies the symmetry of the discretized spatial differential operators, and we employ the partitioned second‐order symplectic Runge–Kutta method to numerically solve the resulted semi‐discrete Hamiltonian ordinary differential equations, which results in fully discretized scheme is symplectic unlike conventional nearly analytic symplectic partitioned Runge–Kutta methods. Theoretical analyses show that the nearly analytic symplectic partitioned Runge–Kutta method based on locally one‐dimensional technique exhibits great higher stability limits and less numerical dispersion than the nearly analytic symplectic partitioned Runge–Kutta method. Numerical experiments are conducted to verify advantages of the nearly analytic symplectic partitioned Runge–Kutta method based on locally one‐dimensional technique, such as their computational efficiency, stability, numerical dispersion and long‐term calculation capability.

Extended multirate infinitesimal step methods: Derivation of order conditions

Multirate methods are specially designed for problems with multiple time scales. The multirate infinitesimal step method (MIS) was developed as a generalization of the so called split-explicit Runge–Kutta methods, where the integration of the fast part is conducted analytically. The MIS method was originally evolved for applications related to numerical weather prediction, i.e. the integration of the compressible Euler equation.In this work, an extension to MIS methods will be presented where an arbitrary Runge–Kutta method (RK) is applied for the integration of the fast component. Furthermore, the order convergence from the original MIS method will be reinvestigated including the derivation of conditions up to order four. Additionally will be presented how well-known methods such as recursive flux splitting multirate method, (Schlegel et al., 2012) partitioned Runge–Kutta method, (Jackiewicz and Vermiglio, 2000) and generalized additive Runge–Kutta method, (Sandu and Günther, 2015) are related to or can be cast as an extended MIS method. An exemplary MIS method of order four with five stages will show that the convergence behaviour not only depends on the applied method for the integration of the fast component. The method will further indicate that the used fast time step plays a significant role.