Rosenbrock Schemes Research Articles

Exponential Integrators for Resistive Magnetohydrodynamics: Matrix-free Leja Interpolation and Efficient Adaptive Time Stepping

Abstract We propose a novel algorithm for the temporal integration of the resistive magnetohydrodynamics (MHD) equations. The approach is based on exponential Rosenbrock schemes in combination with Leja interpolation. It naturally preserves Gauss’s law for magnetism and is unencumbered by the stability constraints observed for explicit methods. Remarkable progress has been achieved in designing exponential integrators and computing the required matrix functions efficiently. However, employing them in MHD simulations of realistic physical scenarios requires a matrix-free implementation. We show how an efficient algorithm based on Leja interpolation that only uses the right-hand side of the differential equation (i.e., matrix free) can be constructed. We further demonstrate that it outperforms Krylov-based exponential integrators as well as explicit and implicit methods using test models of magnetic reconnection and the Kelvin–Helmholtz instability. Furthermore, an adaptive step-size strategy that gives excellent and predictable performance, particularly in the lenient- to intermediate-tolerance regime that is often of importance in practical applications, is employed.

Efficient exponential time integration for simulating nonlinear coupled oscillators

In this paper, we propose an advanced time integration technique associated with explicit exponential Rosenbrock-based methods for the simulation of large stiff systems of nonlinear coupled oscillators. In particular, a novel reformulation of these systems is introduced and a general family of efficient exponential Rosenbrock schemes for simulating the reformulated system is derived. Moreover, we show the required regularity conditions and prove the convergence of these schemes for the system of coupled oscillators. We present an efficient implementation of this new approach and discuss several applications in scientific and visual computing. The accuracy and efficiency of our approach are demonstrated through a broad spectrum of numerical examples, including a nonlinear Fermi–Pasta–Ulam–Tsingou model, elastic and nonelastic deformations as well as collision scenarios focusing on relevant aspects such as stability and energy conservation, large numerical stiffness, high fidelity, and visual accuracy.

Fourth-order two-stage explicit exponential integrators for time-dependent PDEs

Among the family of fourth-order time integration schemes, the two-stage Gauss–Legendre method, which is an implicit Runge–Kutta method based on collocation, is the only superconvergent. The computational cost of this implicit scheme for large systems, however, is very high since it requires solving a nonlinear system at every step. Surprisingly, in this work we show that one can construct and prove convergence results for exponential methods of order four which use two stages only. Specifically, we derive two new fourth-order two-stage exponential Rosenbrock schemes for solving large systems of differential equations. Moreover, since the newly schemes are not only superconvergent but also fully explicit, they turn out to be very competitive compared to the two-stage Gauss–Legendre method as well as other fourth-order time integration schemes. Numerical experiments are given to demonstrate the efficiency of the new integrators.

Numerical Methods of Solving Cauchy Problems with Contrast Structures

Modern numerical methods allowing to solve contrast structure problems in the most efficient way are described. These methods include explicit-implicit Rosenbrock schemes with complex coefficients and fully implicit backward optimal Runge–Kutta schemes. As an integration argument, it is recommended to choose the length of the integral curve arc. This argument provides high reliability of the calculation and sufficiently decreases the complexity of computations for low-order systems. In order to increase the efficiency, we propose an automatic step selection algorithm based on curvature of the integral curve. This algorithm is as efficient as standard algorithms and has sufficiently larger reliability. We show that along with such an automatic step selection it is possible to calculate a posteriori asymptotically precise error estimation. Standard algorithms do not provide such estimations and their actual error quite often exceeds the user-defined tolerance by several orders. The applicability limitations of numerical methods are investigated. In solving superstiff problems, they sometimes do not provide satisfactory results. In such cases, it is recommended to imply approximate analytical methods. Consequently, numerical and analytical methods are complementary.

Linearly implicit Rosenbrock-type Runge–Kutta schemes applied to the Discontinuous Galerkin solution of compressible and incompressible unsteady flows

In this work we investigate the use of linearly implicit Rosenbrock-type Runge–Kutta schemes to integrate in time high-order Discontinuous Galerkin space discretizations of the Navier–Stokes equations. The final goal of this activity is the application of such schemes to the high-order accurate, both in space and time, simulation of turbulent flows. Besides being able to overcome the severe time step restriction of explicit schemes, Rosenbrock schemes have the attractive feature of requiring just one Jacobian matrix evaluation per time step, thus reducing the overall computational effort. Several high-order (up to sixth order) Rosenbrock schemes available in the literature have been implemented and evaluated on benchmark test cases of both compressible and incompressible flows. For the sake of completeness, the sets of coefficients of the schemes here considered have been reported in an Appendix to the paper. An implementation of Rosenbrock schemes for systems of equations with a solution dependent block diagonal matrix multiplying the time derivative is here proposed and described in detail. This can occur, for example, if sets of working variables different from the conservative ones are used in the compressible Navier–Stokes equations. In particular, we have found useful to employ primitive variables based on the logarithms of pressure and temperature in order to ensure the positivity of all thermodynamic variables at the discrete level. The best performing Rosenbrock scheme resulting from our analysis has then been applied to the Implicit Large Eddy Simulation of the transitional flow around the Selig-Donovan SD7003 airfoil.

Rosenbrock methods with an explicit first stage

Traditional Rosenbrock methods suffer from order reduction when applied to partial differential equations with non-homogeneous boundary conditions and source terms. The paper studies a family of Rosenbrock schemes with an explicit first stage. This structure allows one to construct algorithms with high stage orders, and which do not suffer from order reduction. The paper discusses additional order conditions needed for linear stability, for using inexact Jacobians, and implementation aspects. Second- and third-order practical schemes are constructed, and their application to one- and two-dimensional partial differential equations test problems confirm the theoretical findings.

Rosenbrock-type methods applied to discontinuous differential systems

In this paper we study the numerical solution of a discontinuous differential system by a Rosenbrock method. We also focus on one-sided approach in the context of Rosenbrock schemes, and we suggest a technique based on the use of continuous extension, in order to locate the event point, with an application to discontinuous singularly perturbed systems.

A comparison of the efficiency of Rosenbrock and DIRK variants

AbstractWe consider the use of DIRK and Rosenbrock schemes for the time integration of unsteady flow problems. Thereby, two variants of solving the linear systems are compared regarding their efficiency, namely a Jacobian‐free method versus computing an approximate Jacobian. The Rosenbrock schemes are slow in the Jacobian‐free variant, but may be competitive in the other. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

EChem++ – An object-oriented problem solving environment for electrochemistry. Part 7 : Simulation of equilibrium electron transfer processes with implicit Dirichlet boundary conditions

We present a direct approach to include equilibrium electron transfers into electrochemical simulation models based on the finite element method. This reaction step is described by the Nernst equation combined with a mass flux balance equation. Mathematically, the Nernst equation is an implicit Dirichlet boundary condition with two coupled concentrations. These boundary conditions are imposed within a numerical solver consisting of Rosenbrock schemes and an adaptive finite element method by means of Lagrange multipliers. The algorithm described in this study is integrated into the object-oriented, open source code for the problem solving environment EChem++. Simulations of an equilibrium electron transfer under cyclic voltammetric and chronoamperometric conditions in a one-dimensional cell geometry under semi-infinite linear diffusion show the validity and applicability of this method in the context of molecular electrochemistry.

Automatic symbolic derivation of order conditions for two-stage complex Rosenbrock schemes

Rosenbrock methods are known to be good in numerical modeling of stiff problems. In this work, a new symbolic algorithm is proposed for automating the derivation of order conditions for two-stage Rosenbrock schemes with complex coefficients. Software implementation of the proposed algorithm is discussed in detail.

DECOUPLED SCHWARZ ALGORITHMS FOR IMPLICIT DISCRETIZATIONS OF NONLINEAR MONODOMAIN AND BIDOMAIN SYSTEMS

Novel decoupled Schwarz algorithms for the implicit discretizations of the Monodomain and Bidomain systems in three dimensions are constructed and analyzed. Both implicit Euler and linearly implicit Rosenbrock time discretizations are considered. Convergence rate estimates are proven for a domain decomposition preconditioner based on overlapping additive Schwarz techniques and employed in a Newton–Krylov–Schwarz method for the Euler scheme. An analogous result is proven for the same preconditioner applied to the linear systems originated in the Rosenbrock scheme. Several parallel numerical results in three dimensions confirm the convergence rates predicted by the theory and study the performance of our algorithms for a complete heartbeat, for both Bidomain and Monodomain models, Euler and Rosenbrock schemes, fixed time step and adaptive strategies. The results also show the considerable CPU-time reduction of our decoupled Schwarz solvers with respect to fully-implicit Schwarz solvers.

Two-stage complex Rosenbrock schemes for stiff systems

New two-stage Rosenbrock schemes with complex coefficients are proposed for stiff systems of differential equations. The schemes are fourth-order accurate and satisfy enhanced stability requirements. A one-parameter family of L1-stable schemes with coefficients explicitly calculated by formulas involving only fractions and radicals is constructed. A single L2-stable scheme is found in this family. The coefficients of the fourth-order accurate L4-stable scheme previously obtained by P.D Shirkov are refined. Several fourth-order schemes are constructed that are high-order accurate for linear problems and possess the limiting order of L-decay. The schemes proposed are proved to converge. A symbolic computation algorithm is developed that constructs order conditions for multistage Rosenbrock schemes with complex coefficients. This algorithm is used to design the schemes proposed and to obtain fifth-order accurate conditions.

Rosenbrock schemes with complex coefficients for stiff and differential algebraic systems

Many applied problems are described by differential algebraic systems. Complex Rosenbrock schemes are proposed for the numerical integration of differential algebraic systems by the ɛ-embedding method. The method is proved to converge quadratically. The scheme is shown to be applicable even to superstiff systems. The method makes it possible to perform computations with a guaranteed accuracy. An equation is derived that describes the leading term of the error in the method as a function of time. An algorithm extending the method to systems of differential equations for complex-valued functions is proposed. Examples of numerical computations are given.

EChem++ – An object-oriented problem solving environment for electrochemistry: Part 4. Adaptive multilevel finite elements applied to electrochemical models Algorithm and benchmark calculations

We present an approach for the modeling and simulation of electrochemical processes based on a general mathematical formulation, for which we expect future extensibility. For the numerical solution we propose the Rothe method with adaptive time integration performed by Rosenbrock schemes. For spatial integration we choose an adaptive finite element algorithm, controlled by a hierarchical a posteriori error estimator. The algorithm is designed to simulate models of potential-controlled transient experiments in 1D cell geometries and diffusional transport. It is tested for cyclic voltammetry with a fast electron transfer under semi-infinite diffusion conditions. Results of selected Rosenbrock methods are compared. We discuss the software design of the algorithm, as well as its coupling to the electrochemical compiler Ecco within an object-oriented framework. The software developed within this study is part of EChem++ and can be obtained freely as open-source code.

A comparison of time-discretization/linearization approaches for the incompressible Navier–Stokes equations

This paper presents a numerical study of two ways for discretizing and linearizing the time-dependent incompressible Navier–Stokes equations. One approach consists in first applying a semi-discretization in time by a fully implicit θ-scheme. Then, in each discrete time, the equations are linearized by a fixed point iteration. The number of iterations to reach a given stopping criterion is a priori unknown in this approach. In the second approach, Rosenbrock schemes with s stages are used as temporal discretization. The non-linearity of the Navier–Stokes equations is treated internally in the Rosenbrock methods. In each discrete time, exactly s linear systems of equations have to be solved. The numerical study considers five two-dimensional problems with distinct features. Four implicit time stepping schemes and five Rosenbrock methods are involved.

Testing weighted splitting schemes on a one-column transport-chemistry model

In many transport-chemistry models, a huge system of one-dimensional equations of the advection-diffusion-reaction type has to be integrated in time. Typically, this is done with the help of operator splitting. Operator splitting is attractive for complex large-scale transport-chemistry models because it can handle different processes separately in different parts of the computer program. Rosenbrock schemes combined with approximate matrix factorisation (ROS-AMF) are an alternative to operator splitting which does not suffer from splitting errors. However, since the ROS2-AMF schemes are not based on operator splitting, implementation of these methods often requires major changes in the code. In this paper we test another second-order splitting introduced by Strang in 1963, which seemed to be forgotten and rediscovered recently (partially due to its intrinsic parallelism). This splitting, called symmetrically weighted sequential (SWS) splitting, is simple and straightforward to apply, independent of the order of the operators and has an operator-level parallelism. In the experiments, the SWS scheme compares favourably with Strang splitting, but is less accurate than ROS-AMF.

A new approximate matrix factorization for implicit time integration in air pollution modeling

Implicit time stepping typically requires solution of one or several linear systems with a matrix I−τ J per time step where J is the Jacobian matrix. If solution of these systems is expensive, replacing I−τ J with its approximate matrix factorization (AMF) ( I−τ R)( I−τ V) , R+ V= J , often leads to a good compromise between stability and accuracy of the time integration on the one hand and its efficiency on the other hand. For example, in air pollution modeling, AMF has been successfully used in the framework of Rosenbrock schemes. The standard AMF gives an approximation to I−τ J with the error τ 2 RV , which can be significant in norm. In this paper we propose a new AMF. In assumption that − V is an M-matrix, the error of the new AMF can be shown to have an upper bound τ|| R|| , while still being asymptotically O(τ 2) . This new AMF, called AMF+, is equal in costs to standard AMF and, as both analysis and numerical experiments reveal, provides a better accuracy. We also report on our experience with another, cheaper AMF and with AMF-preconditioned GMRES.

Difference schemes for non-autonomous stiff systems of ordinary differential equations

The properties of Rosenbrock schemes with complex coefficients and implicit Runge-Kutta schemes for the numerical solution of Cauchy problems for stiff systems of ordinary differential equations are investigated. The model problem chosen is a linear non-autonomous system. A one-stage second-order approximation Rosenbrock scheme is constructed, which is monotonic and damped to second order when applied to non-autonomous linear systems. This scheme enables one to compute with large step-size outside the boundary layer and can be used for the numerical solution of a broad range of stiff problems which are nearly linear (including, e.g., computations of transients in electrical circuits).