Runge Kutta Chebyshev Research Articles

Third order two-step Runge–Kutta–Chebyshev methods

The well-known high order stabilized codes (such as DUMKA and ROCK) have several drawbacks: numerically obtained stability polynomials (which do not have a closed analytic form), poor internal stability and convergence. RKC-type methods have much better computational properties. However, these types of methods currently have a second order maximum. In this paper, a family of third order stabilized methods with an explicit analytical solution of stability polynomials is presented. This was made possible by usage of two-step Runge–Kutta methods. A new code TSRKC3 is proposed, illustrated by several examples, and compared to existing programs.

A family of two-step second order Runge–Kutta–Chebyshev methods

In this paper, a new family of second order two-step Runge–Kutta–Chebyshev methods is presented. These methods are a generalization of the one-step stabilized methods and have better computational properties compared to them. A new code TSRKC2 is developed and compared to existing solvers at sufficiently large set of examples.

A Fully Explicit Integrator for Modeling Astrophysical Reactive Flows

Simulating complex astrophysical reacting flows is computationally expensive—reactions are stiff and typically require implicit integration methods. The reaction update is often the most expensive part of a simulation, which motivates the exploration of more economical methods. In this research note, we investigate how the explicit Runge–Kutta–Chebyshev (RKC) method performs compared to an implicit method when applied to astrophysical reactive flows. These integrators are applied to simulations of X-ray bursts arising from unstable thermonuclear burning of accreted fuel on the surface of neutron stars. We show that the RKC method performs with similar accuracy to our traditional implicit integrator, but is more computationally efficient when run on CPUs.

Second order stabilized two-step Runge–Kutta methods

Stabilized methods for the numerical solution of ODEs, also called Chebyshev methods, are explicit methods with extended stability domains along the negative real axis. These methods are intended for large mildly stiff problems, originating mainly from parabolic PDEs. In this paper we present explicit two-step Runge–Kutta–Chebyshev methods of order two, which have more than 2.3 times larger stability intervals than the analogous one-step methods. Explicit formulae for stability intervals are derived, as well as an effective recurrent scheme for calculation of methods’ coefficients for arbitrary number of stages. Our numerical experiments confirm the accuracy and stability properties of the proposed methods and show that at least in the case of constant time steps they can compete with the well-known ROCK2 method.

Conservative stabilized Runge-Kutta methods for the Vlasov-Fokker-Planck equation

In this work, we aim at constructing numerical schemes, that are as efficient as possible in terms of cost and conservation of invariants, for the Vlasov–Fokker–Planck system coupled with Poisson or Ampère equation. Splitting methods are used where the linear terms in space are treated by spectral or semi-Lagrangian methods and the nonlinear diffusion in velocity in the collision operator is treated using a stabilized Runge–Kutta–Chebyshev (RKC) integrator, a powerful alternative of implicit schemes. The new schemes are shown to exactly preserve mass and momentum. The conservation of total energy is obtained using a suitable approximation of the electric field. An H-theorem is proved in the semi-discrete case, while the entropy decay is illustrated numerically for the fully discretized problem. Numerical experiments that include investigation of Landau damping phenomenon and bump-on-tail instability are performed to illustrate the efficiency of the new schemes.

A new Runge–Kutta–Chebyshev Galerkin-characteristic finite element method for advection–dispersion problems in anisotropic porous media

We propose a new approach that combines the modified method of characteristics with a unified finite element discretization for the numerical solution of a class of coupled Darcy-advection–dispersion problems in anisotropic porous media. The proposed method benefits from advantages of the method of characteristics in its ability to handle the nonlinear convective terms, while taking advantage of a unified formulation that allows the use of equal-order finite element approximations along with the L2-projection method for all solutions in the problem. In the proposed Galerkin-characteristic finite element framework, the standard Courant–Friedrichs–Lewy condition is relaxed, and time truncation errors are reduced since no stability criterion restricts the choice of time step. For time integration, we use a Runge–Kutta scheme with the Chebyshev polynomials (RKC). The RKC method has an extensive stability field and it is explicit and second-order accurate. In order to assess the quality of the proposed approach, we perform numerical experiments for several categories of test cases. The first category concerns accuracy test examples with known analytical solutions, while the second category focuses on a benchmark problem of miscible flow in a heterogeneous porous medium. The numerical results obtained show high performance and demonstrate the ability of the method to capture dispersion effects in porous media problems.

A fully adaptive explicit stabilized integrator for advection–diffusion–reaction problems

A novel second order family of explicit stabilized Runge–Kutta–Chebyshev methods for advection–diffusion–reaction equations is introduced. The new methods outperform existing schemes for relatively high Peclet number due to their favorable stability properties and explicitly available coefficients. The construction of the new schemes is based on stabilization using second kind Chebyshev polynomials first used in the construction of the stochastic integrator SK-ROCK. An adaptive algorithm to implement the new scheme is proposed. This algorithm is able to automatically select the suitable step size, number of stages, and damping parameter at each integration step. Numerical experiments that illustrate the efficiency of the new algorithm are presented.

Development of time-space adaptive SPH for large deformation and impact

<p indent="0mm">A spatially adaptive smoothed particle hydrodynamics (ASPH) method, where the particles can split and merge according to the adaptive criteria, was developed to improve the computational efficiency of the classical smoothed particle hydrodynamics (SPH) method. Recently, the ASPH algorithm has been widely applied to solve complex problems in mechanics. However, the splitting pattern in the current ASPH algorithm may lead to particle disorder when simulating large deformation and impact problems, and a smaller time step should be performed after particle splitting. An adaptive splitting pattern that is based on a relative position vector and velocity interpolation is introduced in this study. Subsequently, this numerical method is combined with the Runge-Kutta Chebyshev method to achieve adaptive time integration. Thus, the time-space adaptive smoothed particle hydrodynamics method (T-S ASPH) is proposed in this study. This algorithm can increase the computational accuracy by eliminating the particle disorder when simulating the large deformation and impact problems while improving the computational efficiency and time integration accuracy. Typical examples are simulated by the classical SPH, current ASPH and developed T-S ASPH methods. The results indicate that, compared with the other two methods, the proposed T-S ASPH method exhibits higher accuracy, efficiency, stability, and order of convergence for the large deformation and impact problems.

Explicit stabilized multirate method for stiff differential equations

Stabilized Runge–Kutta methods are especially efficient for the numerical solution of large systems of stiff nonlinear differential equations because they are fully explicit. For semi-discrete parabolic problems, for instance, stabilized Runge–Kutta methods overcome the stringent stability condition of standard methods without sacrificing explicitness. However, when stiffness is only induced by a few components, as in the presence of spatially local mesh refinement, their efficiency deteriorates. To remove the crippling effect of a few severely stiff components on the entire system of differential equations, we derive a modified equation, whose stiffness solely depends on the remaining mildly stiff components. By applying stabilized Runge–Kutta methods to this modified equation, we then devise an explicit multirate Runge–Kutta–Chebyshev (mRKC) method whose stability conditions are independent of a few severely stiff components. Stability of the mRKC method is proved for a model problem, whereas its efficiency and usefulness are demonstrated through a series of numerical experiments.

SRKCD: A stabilized Runge–Kutta method for stochastic optimization

We introduce a family of stochastic optimization methods based on the Runge–Kutta–Chebyshev (RKC) schemes. The RKC methods are explicit methods originally designed for solving stiff ordinary differential equations by ensuring that their stability regions are of maximal size. In the optimization context, this allows for larger step sizes (learning rates) and better robustness compared to e.g. the popular stochastic gradient descent method. Our main contribution is a convergence proof for essentially all stochastic Runge–Kutta optimization methods. This shows convergence in expectation with an optimal sublinear rate under standard assumptions of strong convexity and Lipschitz-continuous gradients. For non-convex objectives, we get convergence to zero in expectation of the gradients. The proof requires certain natural conditions on the Runge–Kutta coefficients, and we further demonstrate that the RKC schemes satisfy these. Finally, we illustrate the improved stability properties of the methods in practice by performing numerical experiments on both a small-scale test example and on a problem arising from an image classification application in machine learning.

Stabilized Explicit Integrators for Local Parametrization in Multi-Rigid-Body System Dynamics

Abstract In this work, stabilized explicit integrators for local parametrization are introduced to calculate the dynamics of constrained multi-rigid-body systems, including those based on the orthogonal Runge–Kutta–Chebyshev (RKC) method and the extrapolated stabilized explicit Runge–Kutta (ESERK) method. Both of these methods have large stability regions at the negative real axis, and this property makes them suitable to settle the introduction of the stabilization parameter for a constraint equation. The local vectorial rotation parameters are adopted to describe rotations in each rigid body, and a stabilization technique is developed to transform the differential-algebraic equations (DAE) into a set of first-order ordinary differential equations (ODEs) that can be computed efficiently. Several benchmarks are calculated and the results are compared to those by the generalized-α integrator and ADAMS models, verifying their effectiveness in nonstiff problems.

Mixed-precision explicit stabilized Runge–Kutta methods for single- and multi-scale differential equations

Mixed-precision algorithms combine low- and high-precision computations in order to benefit from the performance gains of reduced-precision without sacrificing accuracy. In this work, we design mixed-precision Runge–Kutta–Chebyshev (RKC) methods, where high precision is used for accuracy, and low precision for stability. Generally speaking, RKC methods are low-order explicit schemes with a stability domain growing quadratically with the number of function evaluations. For this reason, most of the computational effort is spent on stability rather than accuracy purposes. In this paper, we show that a naïve mixed-precision implementation of any Runge–Kutta scheme can harm the convergence order of the method and limit its accuracy, and we introduce a new class of mixed-precision RKC schemes that are instead unaffected by this limiting behavior. We present three mixed-precision schemes: a first- and a second-order RKC method, and a first-order multirate RKC scheme for multiscale problems. These schemes perform only the few function evaluations needed for accuracy (1 or 2 for first- and second-order methods respectively) in high precision, while the rest are performed in low precision. We prove that while these methods are essentially as cheap as their fully low-precision equivalent, they retain the stability and convergence order of their high-precision counterpart. Indeed, numerical experiments confirm that these schemes are as accurate as the corresponding high-precision method.

When and how to split? A comparison of two IMEX splitting techniques for solving advection–diffusion–reaction equations

Many mathematical models of natural phenomena are described by partial differential equations (PDEs) that consist of additive contributions from different physical processes. Classical methods for the numerical solution to such equations are monolithic in that all processes are treated with a single method. Additive numerical methods, in contrast, apply distinct methods to each additive term. There are, however, different ways mathematically to specify the additive terms, and it is not always clear which ways (if any) offer advantages over monolithic methods. This study compares the performance of two different additive splitting techniques (physics-based splitting and dynamic linearization) on a suite of eight test problems that involve advection, reaction, and diffusion with various 2-additive Runge–Kutta methods and (monolithic) Runge–Kutta–Chebyshev (RKC) methods. Results show that dynamic linearization generally outperforms physics-based splitting and so should be preferred as the splitting technique when splitting is required or otherwise desirable. RKC methods are the best performers on three of the eight problems, especially at coarse tolerances, but they can also be prone to severe underperformance.

Synchronization of reaction–diffusion Hopfield neural networks with s-delays through sliding mode control

Synchronization of reaction–diffusion Hopfield neural networks with s-delays via sliding mode control (SMC) is investigated in this paper. To begin with, the system is studied in an abstract Hilbert space C([–r; 0];U) rather than usual Euclid space Rn. Then we prove that the state vector of the drive system synchronizes to that of the response system on the switching surface, which relies on equivalent control. Furthermore, we prove that switching surface is the sliding mode area under SMC. Moreover, SMC controller can also force with any initial state to reach the switching surface within finite time, and the approximating time estimate is given explicitly. These criteria are easy to check and have less restrictions, so they can provide solid theoretical guidance for practical design in the future. Three different novel Lyapunov–Krasovskii functionals are used in corresponding proofs. Meanwhile, some inequalities such as Young inequality, Cauchy inequality,Poincaré inequality, Hanalay inequality are applied in these proofs. Finally, an example is given to illustrate the availability of our theoretical result, and the simulation is also carried out based on Runge–Kutta–Chebyshev method through Matlab.

Magnetic field simulations using explicit time integration with higher order schemes

PurposeA transient magneto-quasistatic vector potential formulation involving nonlinear material is spatially discretized using the finite element method of first and second polynomial order. By applying a generalized Schur complement the resulting system of differential algebraic equations is reformulated into a system of ordinary differential equations (ODE). The ODE system is integrated in time by using explicit time integration schemes. The purpose of this paper is to investigate explicit time integration for eddy current problems with respect to the performance of the first-order explicit Euler scheme and the Runge-Kutta-Chebyshev (RKC) method of higher order.Design/methodology/approachThe ODE system is integrated in time using the explicit Euler scheme, which is conditionally stable by a maximum time step size. To overcome this limit, an explicit multistage RKC time integration method of higher order is used to enlarge the maximum stable time step size. Both time integration methods are compared regarding the overall computational effort.FindingsThe numerical simulations show that a finer spatial discretization forces smaller time step sizes. In comparison to the explicit Euler time integration scheme, the multistage RKC method provides larger stable time step sizes to diminish the overall computation time.Originality/valueThe explicit time integration of the Schur complement vector potential formulation of eddy current problems is accelerated by a multistage RKC method.

Testing an Explicit Method for Multi-compartment Neuron Model Simulation on a GPU

AbstractLarge-scale simulation of multi-compartment models is important for understanding the role of morphological structures of individual neurons for information processing in the brain. In a simulation, partial differential equations (PDEs) that describe the dynamics of neurons have to be solved numerically for each time step. To solve PDEs, numerical methods called implicit methods are used for stability. Implicit methods need to solve simultaneous equations, which can make numerical simulation slow on graphics processing units (GPUs) hardware accelerators for parallel computing. To overcome this problem, we investigated the use of explicit methods for multi-compartment model simulation. We applied the Runge–Kutta–Chebyshev (RKC) method to several cerebellar neuron models including Purkinje cells, granule cells, Golgi cells, and inferior olive cells. Next, we implemented a cerebellar cortical model composed of granule cells, Golgi cells, and Purkinje cells, while using different numerical methods for different cell types. Although explicit methods can be unstable against PDEs, using the RKC method showed sufficient stability for most cases, better computational performance than implicit methods on a GPU, and good reproducibility. In the network simulation, choosing the suitable numerical methods for each cell type achieved faster simulation than that used an implicit method solely. Our results suggest that explicit methods are applicable to multi-compartment models and can accelerate computational speed of simulations. Furthermore, to conduct large-scale simulation of multi-compartment models, choosing efficient numerical methods will be more important.

PRICING AMERICAN OPTIONS WITH THE RUNGE–KUTTA–LEGENDRE FINITE DIFFERENCE SCHEME

This paper presents the Runge–Kutta–Legendre (RKL) finite difference scheme, allowing for an additional shift in its polynomial representation. A short presentation of the stability region, comparatively to the Runge–Kutta–Chebyshev scheme follows. We then explore the problem of pricing American options with the RKL scheme under the one factor Black–Scholes and the two factor Heston stochastic volatility models, as well as the pricing of butterfly spread and digital options under the uncertain volatility model, where a Hamilton–Jacobi–Bellman partial differential equation needs to be solved. We explore the order of convergence in these problems, as well as the option greeks stability, compared to the literature and popular schemes such as Crank–Nicolson, with Rannacher time-stepping.

Development of time-space adaptive smoothed particle hydrodynamics method with Runge-Kutta Chebyshev scheme

Smoothed particle hydrodynamics (SPH) method is a Lagrangian particle method that has been widely used for solving complex mechanics problems. To reduce the SPH computational cost, the adaptive SPH (ASPH) with time-varying particle distribution has been proposed using the particle splitting and merging techniques. However, the particle splitting in the adaptive SPH results in a decreased time step due to the reduced particle spacing and thus requires an increased amount of computing time. In the present work, an explicit Runge-Kutta Chebyshev time stepping scheme is implemented within the adaptive SPH, by including the stabilizing substeps into the Runge-Kutta integration method with the use of the Chebyshev polynomials. The number of stabilizing substeps in a single SPH time step can be adaptive after particle splitting while the Runge-Kutta integration time step remains the same. Therefore, the proposed SPH algorithm is adaptive in both time and space domains. Representative example problems are simulated using the developed time-space adaptive SPH method. It is demonstrated that, as compared to the standard SPH and the conventional adaptive SPH with varying particle distribution only, the presented time-space ASPH method exhibits much higher efficiency without compromising accuracy and the superiority in terms of stability.

Explicit stabilised gradient descent for faster strongly convex optimisation

This paper introduces the Runge–Kutta Chebyshev descent method (RKCD) for strongly convex optimisation problems. This new algorithm is based on explicit stabilised integrators for stiff differential equations, a powerful class of numerical schemes that avoid the severe step size restriction faced by standard explicit integrators. For optimising quadratic and strongly convex functions, this paper proves that RKCD nearly achieves the optimal convergence rate of the conjugate gradient algorithm, and the suboptimality of RKCD diminishes as the condition number of the quadratic function worsens. It is established that this optimal rate is obtained also for a partitioned variant of RKCD applied to perturbations of quadratic functions. In addition, numerical experiments on general strongly convex problems show that RKCD outperforms Nesterov’s accelerated gradient descent.

Improved Runge–Kutta–Chebyshev methods

This study proposes a class of improved Runge–Kutta–Chebyshev (RKC) methods for the stiff systems arising from the spatial discretization of partial differential equations. We can obtain the improved first-order and second-order RKC methods by introducing an appropriate combination technique. The main advantage of our improved RKC methods is that the width of the stability domain along the imaginary axis is significantly increased while the length along the negative real axis has almost no reduction. This implies that our improved RKC methods can extend the application scope of the classical RKC methods. The results of five numerical examples (including the advection–diffusion–reaction equations with dominating advection) show that our improved RKC methods can perform very well.