Additive Runge-Kutta Research Articles

Preconditioned Smoothers for the Full Approximation Scheme for the RANS Equations

We consider multigrid methods for finite volume discretizations of the Reynolds averaged Navier–Stokes equations for both steady and unsteady flows. We analyze the effect of different smoothers based on pseudo time iterations, such as explicit and additive Runge–Kutta (AERK) methods. Furthermore, by deriving them from Rosenbrock smoothers, we identify some existing schemes as a class of additive W (AW) methods. This gives rise to two classes of preconditioned smoothers, preconditioned AERK and AW, which are implemented the exact same way, but have different parameters and properties. This derivation allows to choose some of these based on results for time integration methods. As preconditioners, we consider SGS preconditioners based on flux vector splitting discretizations with a cutoff function for small eigenvalues. We compare these methods based on a discrete Fourier analysis. Numerical results on pitching and plunging airfoils identify AW3 as the best smoother regarding overall efficiency. Specifically, for the NACA 64A010 airfoil steady-state convergence rates of as low as 0.85 were achieved, or a reduction of 6 orders of magnitude in approximately 25 pseudo-time iterations. Unsteady convergence rates of as low as 0.77 were achieved, or a reduction of 11 orders of magnitude in approximately 70 pseudo-time iterations.

Stability Analysis of Additive Runge-Kutta Methods for Delay-Integro-Differential Equations

This paper is concerned with stability analysis of additive Runge-Kutta methods for delay-integro-differential equations. We show that if the additive Runge-Kutta methods are algebraically stable, the perturbations of the numerical solutions are controlled by the initial perturbations from the system and the methods.

Kinetic Simulation of Collisional Magnetized Plasmas with Semi-implicit Time Integration

Plasmas with varying collisionalities occur in many applications, such as tokamak edge regions, where the flows are characterized by significant variations in density and temperature. While a kinetic model is necessary for weakly-collisional high-temperature plasmas, high collisionality in colder regions render the equations numerically stiff due to disparate time scales. In this paper, we propose an implicit–explicit algorithm for such cases, where the collisional term is integrated implicitly in time, while the advective term is integrated explicitly in time, thus allowing time step sizes that are comparable to the advective time scales. This partitioning results in a more efficient algorithm than those using explicit time integrators, where the time step sizes are constrained by the stiff collisional time scales. We implement semi-implicit additive Runge–Kutta methods in COGENT, a high-order finite-volume gyrokinetic code and test the accuracy, convergence, and computational cost of these semi-implicit methods for test cases with highly-collisional plasmas.

Implicit–explicit (IMEX) Runge–Kutta methods for non-hydrostatic atmospheric models

Abstract. The efficient simulation of non-hydrostatic atmospheric dynamics requires time integration methods capable of overcoming the explicit stability constraints on time step size arising from acoustic waves. In this work, we investigate various implicit–explicit (IMEX) additive Runge–Kutta (ARK) methods for evolving acoustic waves implicitly to enable larger time step sizes in a global non-hydrostatic atmospheric model. The IMEX formulations considered include horizontally explicit – vertically implicit (HEVI) approaches as well as splittings that treat some horizontal dynamics implicitly. In each case, the impact of solving nonlinear systems in each implicit ARK stage in a linearly implicit fashion is also explored.The accuracy and efficiency of the IMEX splittings, ARK methods, and solver options are evaluated on a gravity wave and baroclinic wave test case. HEVI splittings that treat some vertical dynamics explicitly do not show a benefit in solution quality or run time over the most implicit HEVI formulation. While splittings that implicitly evolve some horizontal dynamics increase the maximum stable step size of a method, the gains are insufficient to overcome the additional cost of solving a globally coupled system. Solving implicit stage systems in a linearly implicit manner limits the solver cost but this is offset by a reduction in step size to achieve the desired accuracy for some methods. Overall, the third-order ARS343 and ARK324 methods performed the best, followed by the second-order ARS232 and ARK232 methods.

Exponential Additive Runge-Kutta Methods for Semi-Linear Differential Equations

AbstractExponential additive Runge-Kutta methods for solving semi-linear equations are discussed. Related order conditions and stability properties for both explicit and implicit schemes are developed, according to the dimension of the coefficients in the linear terms. Several examples illustrate our theoretical results.

Implementation of a Semi-implicit Time Integration Scheme in Non-Hydrostatic Euler Equations

A semi-implicit time integration scheme is implemented in a non-hydrostatic Euler problem on the cubed-sphere grid. The semi-implicit time integration scheme is a 3-stage additive RungeKutta method, which is an Implicit-Explicit (IMEX) multi-stage single-step scheme. The system treats acoustic and gravity waves implicitly and advection explicitly. In the implicit part, we compute the linear system by defining a proper linear operator. Then, the numerical results are presented to compare the semi-implicit time integration scheme with the explicit RungeKutta time integration scheme in non-hydrostatic Euler equations. In terms of accuracy, we demonstrate that the proposed method performs better than other schemes.

Multirate finite step methods for linear acoustics

AbstractMany processes contain phenomena on different time scales, leading to model equations with fast and small parts. There are several approaches to solve these equations, like additive Runge Kutta methods or multirate infinitesimal steps methods (MIS). Both methods make use of the additive splitting of the ODE in fast and small parts. The multiple infinitesimal step method integrates the slow part with a large macro stepsize, whereas the fast terms are solved with several smaller steps of a simpler method. The order conditions of a MIS method are derived under the assumption of the exact integration of the fast parts.We develop the multirate finite step methods (MFS). These methods are derived from the MIS methods, by taking a simple small scale integrator for the fast terms. This small scale integrator uses the same number of steps in each stage. With these assumptions, we derive the order conditions, such that the order is independent in the number of small steps. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Semi-implicit Time Integration of Atmospheric Flows with Characteristic-Based Flux Partitioning

This paper presents a characteristic-based flux partitioning for the semi-implicit time integration of atmospheric flows. Nonhydrostatic models require the solution of the compressible Euler equations. The acoustic time-scale is significantly faster than the advective scale, yet it is typically not relevant to atmospheric and weather phenomena. The acoustic and advective components of the hyperbolic flux are separated in the characteristic space. High-order, conservative additive Runge-Kutta methods are applied to the partitioned equations so that the acoustic component is integrated in time implicitly with an unconditionally stable method, while the advective component is integrated explicitly. The time step of the overall algorithm is thus determined by the advective scale. Benchmark flow problems are used to demonstrate the accuracy, stability, and convergence of the proposed algorithm. The computational cost of the partitioned semi-implicit approach is compared with that of explicit time integration.

Multirate generalized additive Runge Kutta methods

This work constructs a new class of multirate schemes based on the recently developed generalized additive Runge---Kutta (GARK) methods (Sandu and Gunther, SIAM J Numer Anal, 53(1):17---42, 2015). Multirate schemes use different step sizes for different components and for different partitions of the right-hand side based on the local activity levels. We show that the new multirate GARK family includes many well-known multirate schemes as special cases. The order conditions theory follows directly from the GARK accuracy theory. Nonlinear stability and monotonicity investigations show that these properties are inherited from the base schemes provided that additional coupling conditions hold.

Numerical simulation of H2–O2 gaseous detonation on the wedge

2D cellular reflection processes of detonation waves on different angle wedges were investigated by using a fifth-order weighted essentially non-oscillatory (WENO) scheme with Additive Runge–Kutta(ARK) time step method. A detailed chemical kinetics model was adopted to construct the mass fraction of species in a stoichiometric H2–O2 mixture diluted by Argon. It was found that a Mach reflection as well as a regular reflection existed over the wedge. The regular cellular detonation wave was disturbed where the Mach reflection occurs, which made the cells become smaller and the shape get distorted. It was shown from the numerical results that when the angle is less than 30°, the trajectory of the triple points was not linearly distributed. However, it got close to be a linear distribution with the increase of the angle. With the increase of wedge angle, the pressure on the wedge surface got increased especially for the apex which suffered a higher pressure. The state of the chemical reaction was indicated by the intermediate concentration in which the incident shock was weaker than the Mach stem. These numerical results and conclusions were significant for the evaluation of detonation threats and meaningful in engineering application.

Nonlinear Stability and B-convergence of Additive Runge-Kutta Methods for Nonlinear Stiff Problems

AbstractIn this paper, we are devoted to nonlinear stability and B-convergence of additive Runge-Kutta (ARK) methods for nonlinear stiff problems with multiple stiffness. The concept of -algebraic stability of ARK methods for a class of stiff problems Kστ is introduced, and it is proven that this stability implies some contractive properties of the ARK methods. Some results on optimal B-convergence of ARK methods for Kσ,0 are given. These new results extend the existing ones of RK methods and ARK methods in the references. Numerical examples test the correctness of our theoretical analysis.

Fast Solver for the Local Discontinuous Galerkin Discretization of the KdV Type Equations

AbstractIn this paper, we will develop a fast iterative solver for the system of linear equations arising from the local discontinuous Galerkin (LDG) spatial discretization and additive Runge-Kutta (ARK) time marching method for the KdV type equations. Being implicit in time, the severe time step , with the k-th order of the partial differential equations (PDEs)) restriction for explicit methods will be removed. The equations at the implicit time level are linear and we demonstrate an efficient, practical multigrid (MG) method for solving the equations. In particular, we numerically show the optimal or sub-optimal complexity of the MG solver and a two-level local mode analysis is used to analyze the convergence behavior of the MG method. Numerical results for one-dimensional, two-dimensional and three-dimensional cases are given to illustrate the efficiency and capability of the LDG method coupled with the multigrid method for solving the KdV type equations.

A fourth-order approximate projection method for the incompressible Navier–Stokes equations on locally-refined periodic domains

In this follow-up of our previous work [30], the author proposes a high-order semi-implicit method for numerically solving the incompressible Navier–Stokes equations on locally-refined periodic domains. Fourth-order finite-volume stencils are employed for spatially discretizing various operators in the context of structured adaptive mesh refinement (AMR). Time integration adopts a fourth-order, semi-implicit, additive Runge–Kutta method to treat the non-stiff convection term explicitly and the stiff diffusion term implicitly. The divergence-free condition is fulfilled by an approximate projection operator. Altogether, these components yield a simple algorithm for simulating incompressible viscous flows on periodic domains with fourth-order accuracies both in time and in space. Results of numerical tests show that the proposed method is superior to previous second-order methods in terms of accuracy and efficiency. A major contribution of this work is the analysis of a fourth-order approximate projection operator.

Numerical application of additive Runge-Kutta methods on detonation interaction with pipe bends

A detailed reaction model comprised of 9 species and 48 reactions is employed in simulating two-dimensional cellular detonations propagating through smooth pipe bends in a stoichiometric H2/O2 mixture diluted by Argon. Additive Runge-Kutta (ARK) methods are applied to solve the stiff reactive Euler equations, in which the stiff and non-stiff terms are solved implicitly and explicitly. The numerical results indicate that, as the regular cellular detonation wave propagating through the bend section, the diffraction near the inner wall causes an increase in detonation cell size while the detonation reflection occurring on the bottom wall leads to a decrease in cell size. In addition, an expansion wave is generated continuously. The expansion wave causes the failure as well as the partial failure of the detonations near the inner and outer walls, respectively. On the contrary, the transverse re-initiation waves evolve into a detonation in the decoupling zone just downstream of the bend outlet owing to continuous compression imposed by other transverse waves propagating right behind. Meanwhile, there exists a transition length after the detonation propagating out of the bend and entering the sloped tube section.

Additive Runge-Kutta methods for H2/O2/Ar detonation with a detailed elementary chemical reaction model

We report here the additive Runge-Kutta methods for computing reactive Euler equations with a stiff source term, and in particular, their applications in gaseous detonation simulations. The source term in gaseous detonation is stiff due to the presence of wide range of time scales during thermal-chemical non-equilibrium reactive processes and some of these time scales are much smaller than that of hydrodynamic flow. The high order, L-stable, additive Runge-Kutta methods proposed in this paper resolved the stiff source term into the stiff part and non-stiff part, in which the stiff part was solved implicitly while the non-stiff part was handled explicitly. The proposed method was successfully applied to simulating the gaseous detonation in a stoichiometric H2/O2/Ar mixture based on a detailed elementary chemical reaction model comprised of 9 species and 19 elementary reactions. The results showed that the stiffly accurate additive Runge-Kutta methods can capture the discontinuity well, and describe the detonation complex wave configurations accurately such as the triple wave structure and cellular pattern.

Energy-conserving Runge–Kutta methods for the incompressible Navier–Stokes equations

Energy-conserving methods have recently gained popularity for the spatial discretization of the incompressible Navier–Stokes equations. In this paper implicit Runge–Kutta methods are investigated which keep this property when integrating in time. Firstly, a number of energy-conserving Runge–Kutta methods based on Gauss, Radau and Lobatto quadrature are constructed. These methods are suitable for convection-dominated problems (such as turbulent flows), because they do not introduce artificial diffusion and are stable for any time step. Secondly, to obtain robust time-integration methods that work also for stiff problems, the energy-conserving methods are extended to a new class of additive Runge–Kutta methods, which combine energy conservation with L-stability. In this class, the Radau IIA/B method has the best properties. Results for a number of test cases on two-stage methods indicate that for pure convection problems the additive Radau IIA/B method is competitive with the Gauss methods. However, for stiff problems, such as convection-dominated flows with thin boundary layers, both the higher order Gauss and Radau IIA/B method suffer from order reduction. Overall, the Gauss methods are the preferred method for energy-conserving time integration of the incompressible Navier–Stokes equations.

Variable step-size fractional step Runge–Kutta methods for time-dependent partial differential equations

Fractional step Runge–Kutta methods are a class of additive Runge–Kutta schemes that provide efficient time discretizations for evolutionary partial differential equations. This efficiency is due to appropriate decompositions of the elliptic operator involving the spatial derivatives. In this work, we tackle the design and analysis of embedded pairs of fractional step Runge–Kutta methods. Such methods suitably estimate the local error at each time step, thus providing efficient variable step-size time integrations. Finally, some numerical experiments illustrate the behaviour of the proposed algorithms.

Semi-implicit integral deferred correction constructed with additive Runge–Kutta methods

In this paper, we consider construct high order semi-implicit integrators using integral deferred correction (IDC) to solve stiff initial value problems. The general framework for the construction of these semi-implicit methods uses uniformly distributed nodes and additive RungeKutta (ARK) integrators as base schemes inside an IDC framework, which we refer to as IDC-ARK methods. We establish under mild assumptions that, when an r order ARK method is used to predict and correct the numerical solution, the order of accuracy of the IDC method increases by r for each IDC prediction and correction loop. Numerical experiments support the established theorems, and also indicate that higher order IDC-ARK methods present an efficiency advantage over existing implicit-explicit (IMEX) ARK schemes in some cases.

Split Newton iterative algorithm and its application

Inspired by some implicit–explicit linear multistep schemes and additive Runge–Kutta methods, we develop a novel split Newton iterative algorithm for the numerical solution of nonlinear equations. The proposed method improves computational efficiency by reducing the computational cost of the Jacobian matrix. Consistency and global convergence of the new method are also maintained. To test its effectiveness, we apply the method to nonlinear reaction–diffusion equations, such as Burger’s–Huxley equation and fisher’s equation. Numerical examples suggest that the involved iterative method is much faster than the classical Newton’s method on a given time interval.

Numerical Stability Study of the Explicit-Implicit Range-Kuta Synthesis Methods (IMEX-ARK)

In this research we studied the numerical stability for Implicit-Explicit Additive Runge-Kutta (IMEX-ARK) methods which have the A[]-stability .This stability is equivalent A-stability of (B()-ARK) methods where B()=B1+(1-)B2 of second and third order for different intervals of values.This methods are suitable for solving stiff non-linear differential equations which has the form : which contains stiff and non-stiff terms. We have used Matlab as programming tool .