Diagonal Implicit Runge-Kutta Methods Research Articles

Construction of A-stable explicit last-stage diagonal implicit Runge–Kutta (ELDIRK) methods

AbstractELDIRK methods are defined to have an Explicit Last stage in the general Butcher array of Diagonal Implicit Runge-Kutta methods, with the consequence, that no additional system of equations must be solved, compared to the embedded RK method. Two general formulations for second- and third-order ELDIRK methods have been obtained recently in Mahnken [21] with specific schemes, e.g. for the embedded implicit Euler method, the embedded trapezoidal-rule and the embedded Ellsiepen method. In the first part of this paper, we investigate some general stability characteristics of ELDIRK methods, and it will be shown that the above specific RK schemes are not A-stable. Therefore, in the second part, the above-mentioned general formulations are used for further stability investigations, with the aim to construct new second- and third-order ELDIRK methods which simultaneously are A-stable. Two numerical examples are concerned with the curing for a thermosetting material and phase-field RVE modeling for crystallinity and orientation. The numerical results confirm the theoretical results on convergence order and stability.

Accuracy and Stability Analysis of the Semi-Lagrangian Method for Stiff Hyperbolic Relaxation Systems and Kinetic BGK Model

In this paper, we develop a family of third order asymptotic-preserving (AP) and asymptotically accurate (AA) diagonally implicit Runge–Kutta (DIRK) time discretization methods for the stiff hyperbolic relaxation systems and kinetic Bhatnagar–Gross–Krook (BGK) model in the semi-Lagrangian (SL) setting. The methods are constructed based on an accuracy analysis of the SL scheme for stiff hyperbolic relaxation systems and kinetic BGK model in the limiting fluid regime when the Knudsen number approaches 0. An extra order condition for the asymptotic third order accuracy in the limiting regime is derived. Linear von Neumann stability analysis of the proposed third order DIRK methods are performed to a simplified two-velocity linear kinetic model. Extensive numerical tests are presented to demonstrate the AA, AP, and stability properties of our proposed schemes.

High-order bicompact schemes for the quasilinear multidimensional diffusion equation

High-order bicompact schemes are designed for the quasilinear multidimensional diffusion equation. They are constructed using the method of lines, the finite volume method, the bicubic (or tricubic) Hermite interpolation. Implicit-explicit and diagonally-implicit Runge-Kutta methods are applied to time integration. The resulting schemes are stable for any ratios between grid steps, are conservative, have approximation of the fourth order in space and the third order in time. To implement the implicit-explicit schemes, an iteration method is proposed based on the approximate factorization of their multidimensional difference operators. This method is modified to implement the schemes with diagonally-implicit Runge-Kutta time stepping. High-order grid convergence and implementation efficiency of the new bicompact schemes are demonstrated on numerical examples.

Comparison of implicit time-discretization schemes for hybridized discontinuous Galerkin methods

The present study is focused on the application of two families of implicit time-integration schemes for general time-dependent balance laws of convection-diffusion-reaction type discretized by a hybridized discontinuous Galerkin method in space, namely backward differentiation formulas (BDF) and diagonally implicit Runge-Kutta (DIRK) methods. Special attention is devoted to embedded DIRK methods, which allow the incorporation of time step size adaptation algorithms in order to keep the computational effort as low as possible. The properties of the numerical solution, such as its order of convergence, are investigated by means of suitably chosen test cases for a linear convection-diffusion-reaction equation and the nonlinear system of Navier-Stokes equations. For problems considered in this work, the DIRK methods prove to be superior to high-order BDF methods in terms of both stability and accuracy.

Simulating the temporal change of the active response of arteries by finite elements with high-order time-integrators

A new numerical method to model the active response of arteries is proposed. Vasoconstrictors and vasodilators in the bloodstream diffuse from the lumen into the arterial wall through the intima and cause the smooth muscle cells, mostly in the media, to contract. We combine the diffusion process with the mechanical model in Yosibash and Priel (Comput Methods Appl Mech Eng 237–240:51–66, 2012). Finite element computations of the fully coupled field problem using time-adaptive, high-order time-integration methods based on diagonally-implicit Runge–Kutta methods are investigated with respect to their convergence behavior for linear and non-linear loading paths. Since the blood pressure is periodic, highly non-linear external loading path, the step-size estimation has to be adapted to minimize step-size rejections. An example of an artery analysis that illustrates the advantage of the proposed time-adaptive scheme is provided.

Exponentially fitted symmetric and symplectic DIRK methods for oscillatory Hamiltonian systems

The order conditions for modified Runge–Kutta methods are derived via the rooted trees. Symmetry and symplecticity conditions and exponential fitting conditions for modified diagonally implicit Runge–Kutta (DIRK) are considered. Three new exponentially fitted symmetric and symplectic diagonally implicit Runge–Kutta (EFSSDIRK) methods of respective second order and fourth order are constructed. Phase properties of the new methods are analyzed. The new EFSSDIRK methods are applied to several Hamiltonian problems and compared to the results obtained by the existing symplectic DIRK methods in the literature.

Order Barrier for Low-Storage DIRK Methods with Positive Weights

In this paper we study an order barrier for low-storage diagonally implicit Runge–Kutta (DIRK) methods with positive weights. The Butcher matrix for these schemes, that can be implemented with only two memory registers in the van der Houwen implementation, has a special structure that restricts the number of free parameters of the method. We prove that third order low-storage DIRK methods must contain negative weights, obtaining the order barrier \(p\le 2\) for these schemes. This result extends the well known one for symplectic DIRK methods, which are a particular case of low-storage DIRK methods. Some other properties of second order low-storage DIRK methods are given.

Stage-parallel fully implicit Runge–Kutta solvers for discontinuous Galerkin fluid simulations

In this paper, we develop new techniques for solving the large, coupled linear systems that arise from fully implicit Runge–Kutta methods. This method makes use of the iterative preconditioned GMRES algorithm for solving the linear systems, which has seen success for fluid flow problems and discontinuous Galerkin discretizations. By transforming the resulting linear system of equations, one can obtain a method which is much less computationally expensive than the untransformed formulation, and which compares competitively with other time-integration schemes, such as diagonally implicit Runge–Kutta (DIRK) methods. We develop and test several ILU-based preconditioners effective for these large systems. We additionally employ a parallel-in-time strategy to compute the Runge–Kutta stages simultaneously. Numerical experiments are performed on the Navier–Stokes equations using Euler vortex and 2D and 3D NACA airfoil test cases in serial and in parallel settings. The fully implicit Radau IIA Runge–Kutta methods compare favorably with equal-order DIRK methods in terms of accuracy, number of GMRES iterations, number of matrix–vector multiplications, and wall-clock time, for a wide range of time steps.

High order semi-Lagrangian methods for the BGK equation

A new class of high-order accuracy numerical methods for the BGK model of the Boltzmann equation is presented. The schemes are based on a semi-lagrangian formulation of the BGK equation; time integration is dealt with DIRK (Diagonally Implicit Runge Kutta) and BDF methods; the latter turn out to be accurate and computationally less expensive than the former. Numerical results and examples show that the schemes are reliable and efficient for the investigation of both rarefied and fluid regimes in gasdynamics.

A monolithic finite element approach using high-order schemes in time and space applied to finite strain thermo-viscoelasticity

This article addresses a thermo-mechanically coupled problem of thermo-viscoelasticity at finite strains using a monolithic approach. The underlying equations are based on the non-linear transient heat equation, the local equilibrium conditions and the evolution equations of the internal variables. The latter describe the hardening behavior of the material. If the method of vertical lines is applied, its first step–namely the spatial discretization–yields a system of differential-algebraic equations (DAE-system). Here, we employ the p-version of the finite element method based on integrated Legendre polynomials. This can lead to very precise solutions in the spatial domain. In order to be accurate in the time-domain as well, stiffly accurate, diagonally-implicit Runge–Kutta methods are applied to solve the DAE-system yielding a coupled system of non-linear algebraic equations. In this article, the system is solved monolithically by employing the Multilevel-Newton algorithm. Accordingly, a high-order result is obtained in the space and the time domain. The numerical concept is applied to a constitutive model of finite strain thermo-viscoelasticity. Several examples are applied to demonstrate the efficiency and applicability of the numerical scheme. It is especially the transient problems that call for time-adaptive schemes which are naturally embedded in the concept.

Steady-State Algorithms for Nonlinear Time-Periodic Magnetic Diffusion Problems Using Diagonally Implicit Runge–Kutta Methods

A general framework is presented for the formulation of steady-state simulation algorithms for magnetically nonlinear eddy-current problems using implicit Runge-Kutta (RK) methods. A close analogy is drawn between equations discretized using the backward-Euler method and fully implicit RK methods. Detailed formulations of the time-periodic finite-element method (TPFEM) and the shooting-Newton method (SNM) are derived using the popular family of diagonally implicit RK (DIRK) methods. Both algorithms employ the generalized minimum residual method to solve the linear equations arising at each Newton iteration. The benefits of higher-order DIRK methods are demonstrated by simulating a surface mount permanent magnet synchronous machine. The effects of using a solid versus a laminated rotor back iron on the simulation time are examined. Simulation results indicate that the performance of TPFEM and SNM is quite similar and much faster than transient analysis.

Numerical solutions of systems with (p,δ)‐structure using local discontinuous Galerkin finite element methods

SUMMARYIn this paper, we present LDG methods for systems with (p,δ)‐structure. The unknown gradient and the nonlinear diffusivity function are introduced as auxiliary variables and the original (p,δ) system is decomposed into a first‐order system. Every equation of the produced first‐order system is discretized in the discontinuous Galerkin framework, where two different nonlinear viscous numerical fluxes are implemented. An a priori bound for a simplified problem is derived. The ODE system resulting from the LDG discretization is solved by diagonal implicit Runge–Kutta methods. The nonlinear system of algebraic equations with unknowns the intermediate solutions of the Runge–Kutta cycle is solved using Newton and Picard iterative methodology. The performance of the two nonlinear solvers is compared with simple test problems. Numerical tests concerning problems with exact solutions are performed in order to validate the theoretical spatial accuracy of the proposed method. Further, more realistic numerical examples are solved in domains with non‐smooth boundary to test the efficiency of the method. Copyright © 2014 John Wiley & Sons, Ltd.

A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows

The potential of the hybridized discontinuous Galerkin (HDG) method has been recognized for the computation of stationary flows. Extending the method to time-dependent problems can, e.g., be done by backward differentiation formula (BDF) or diagonally implicit Runge–Kutta (DIRK) methods. In this work, we investigate the use of embedded DIRK methods in an HDG solver, including the use of adaptive time-step control. Numerical results demonstrate the performance of the method for both linear and nonlinear (systems of) time-dependent convection–diffusion equations.

A stiffly accurate Rosenbrock-type method of order 2 applied to FE-analyses in finite strain viscoelasticity

The numerical treatment of quasi-static problems in solid mechanics incorporating constitutive equations of evolutionary type is pursued using the method of vertical lines. The spatial discretization is carried out by means of the finite element method. This yields large systems of differential-algebraic equations. In this article a new stiffly accurate Rosenbrock-type method of second order is proposed, which is applied to this challenging test case, and comparisons to existing Rosenbrock-type and diagonally-implicit Runge–Kutta methods are carried out.

Implicit peer methods for large stiff ODE systems

Implicit two-step peer methods are introduced for the solution of large stiff systems. Although these methods compute s-stage approximations in each time step one-by-one like diagonally-implicit Runge-Kutta methods the order of all stages is the same due to the two-step structure. The nonlinear stage equations are solved by an inexact Newton method using the Krylov solver FOM (Arnoldi’s method). The methods are zero-stable for arbitrary step size sequences. We construct different methods having order p=s in the multi-implicit case and order p=s−1 in the singly-implicit case with arbitrary step sizes and s≤5. Numerical tests in Matlab for several semi-discretized partial differential equations show the efficiency of the methods compared to other Krylov codes.

Adaptive time step control for the incompressible Navier–Stokes equations

Adaptive time stepping is an important tool in Computational Fluid Dynamics for controlling the accuracy of simulations and for enhancing their efficiency. This paper presents a systematic study of three classes of implicit and linearly implicit time stepping schemes with adaptive time step control applied to a 2D laminar flow around a cylinder: θ -schemes, diagonal-implicit Runge–Kutta (DIRK) methods and Rosenbrock–Wanner (ROW) methods. The time step is controlled using embedded methods. It is shown that several ROW methods clearly outperform the more standard θ -schemes and the DIRK methods. The results depend on a prescribed tolerance in the time step control algorithm, whose appropriate choice varies from scheme to scheme.

Stability of explicit and diagonal implicit Runge–Kutta methods for nonlinear Volterra functional differential equations in Banach spaces

This paper is concerned with the stability properties of explicit and diagonal implicit Runge–Kutta methods when applied to initial value problems of nonlinear Volterra functional differential equations in Banach spaces. A series of new stability results of explicit and diagonal implicit Runge–Kutta methods are obtained.

Linear-implicit time integration schemes for error-controlled transient nonlinear magnetic field simulations

Linear-implicit time-step methods of the Rosenbrock type are proposed for the time integration of the nonlinear differential-algebraic systems of equations that arise in transient magnetic field simulations. These methods avoid the iterative solution of the nonlinear systems due to their built-in Newton procedures. Embedded lower order schemes allow an error-controlled adaptive time stepping to take into account the dynamics of the underlying process. Numerical tests show the applicability of these methods for error-controlled adaptive time integration of nonlinear magnetodynamic problems and the comparison to established singly diagonal implicit Runge-Kutta methods shows the benefits and possible problems of these specialized Runge-Kutta methods.

Applying Higher Order DIRK Time Steps to the “Modified Picard” Method

AbstractThe “modified Picard” iteration method, which offers global mass conservation, can also be described as a form of Newton's iteration with lagged nonlinear coefficients. It converges to a time step with first‐order discretization error. This paper applies second‐ and third‐order diagonally implicit Runge Kutta (DIRK) time steps to the modified Picard method in one example. It demonstrates improvements over the first‐order time step in rms error and error‐times‐effort model quality by factors ranging from two to over two orders of magnitude, showing that the “modified Picard” and DIRK methods are compatible.

Poisson schemes for Hamiltonian systems on Poisson manifolds

In this paper, we construct Poisson difference schemes of any order accuracy based on Padé approximation for linear Hamiltonian systems on Poisson manifolds with constant coefficients. For nonlinear Hamiltonian systems on Poisson manifolds, we point out that symplectic diagonal implicit Runge-Kutta methods are also Poisson schemes. The preservation of distinguished functions and quadratic first integrals of the original Hamiltonian systems of these schemes are also discussed.