Rosenbrock-type Methods Research Articles

Rosenbrock-Type Methods for Solving Stochastic Differential Equations

Rosenbrock-Type Methods for Solving Stochastic Differential Equations

Weak Convergence of the Rosenbrock Semi-implicit Method for Semilinear Parabolic SPDEs Driven by Additive Noise

Abstract This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability properties. Exponential Rosenbrock and Rosenbrock-type methods were proved to be efficient for such SPDEs, but only their strong convergence were recently analyzed. Here, we investigate the weak convergence of the Rosenbrock semi-implicit method. We obtain a weak convergence rate which is twice the rate of the strong convergence. Our error analysis does not rely on Malliavin calculus, but rather only uses the Kolmogorov equation and the smoothing properties of the resolvent operator resulting from the Rosenbrock semi-implicit approximation.

Rosenbrock–Wanner and W-methods for the Navier–Stokes equations

This paper presents a numerical study of using Rosenbrock-type methods for the temporal discretization of the incompressible Navier–Stokes equations. We focus on Rosenbrock–Wanner (ROW) and W-methods. In these methods, the treatment of the Navier–Stokes nonlinearity is built directly into the time-stepping schemes. Hence, in each time step, s merely linear systems have to be solved, where s is the number of stages of these methods.While in ROW methods the system matrix is different in each time step, W-methods grant the possibility of keeping the system matrix fixed for some or even all time steps.In this paper, we conduct numerical studies in which we examine four different Rosenbrock-type methods in their ROW method variants and several different W-method variants. We study a manufactured example with known solution and the well-known benchmark problem “flow around a circular obstacle”.

Rosenbrock Type Methods for Solving Non-Linear Second-Order in Time Problems

In this work, we develop a new class of methods which have been created in order to numerically solve non-linear second-order in time problems in an efficient way. These methods are of the Rosenbrock type, and they can be seen as a generalization of these methods when they are applied to second-order in time problems which have been previously transformed into first-order in time problems. As they also follow the ideas of Runge–Kutta–Nyström methods when solving second-order in time problems, we have called them Rosenbrock–Nyström methods. When solving non-linear problems, Rosenbrock–Nyström methods present less computational cost than implicit Runge–Kutta–Nyström ones, as the non-linear systems which arise at every intermediate stage when Runge–Kutta–Nyström methods are used are replaced with sequences of linear ones.

Schemes of (m, k)-Type for Solving Differential-Algebraic and Stiff Systems

A form of Rosenbrock-type methods optimal in terms of the number of non-zero parameters and computational costs per step is considered. A technique of obtaining $$(m, k)$$-methods from some well-known Rosenbrock-type methods is justified. Formulas for transforming the parameters of $$(m,k)$$-schemes and for obtaining a stability function are given for two canonical representations of the schemes. An $$L$$-stable $$(3, 2)$$-method of order 3 is proposed, which requires two evaluations of the function: one evaluation of the Jacobian matrix and one $$LU$$-decomposition per step. A variable step size integration algorithm based on the $$(3,2)$$-method is formulated. It provides a numerical solution for both explicit and implicit systems of ODEs. Numerical results are presented to show the efficiency of the new algorithm.

Study on split-step Rosenbrock type method for stiff stochastic differential systems

ABSTRACT In this paper, we present a split-step Rosenbrock type method for stiff stochastic differential systems. The method is proved to be mean-square (MS) convergent with strong order . For one- and two-dimensional Itô test equations with multiplicative noise, we analysis asymptotically MS stability and plot asymptotic MS stability regions. Numerical examples and simulations are given to illustrate the effectiveness of theoretical results.

A Robust and Efficient Numerical Method for RNA-Mediated Viral Dynamics.

The multiscale model of hepatitis C virus (HCV) dynamics, which includes intracellular viral RNA (vRNA) replication, has been formulated in recent years in order to provide a new conceptual framework for understanding the mechanism of action of a variety of agents for the treatment of HCV. We present a robust and efficient numerical method that belongs to the family of adaptive stepsize methods and is implicit, a Rosenbrock type method that is highly suited to solve this problem. We provide a Graphical User Interface that applies this method and is useful for simulating viral dynamics during treatment with anti-HCV agents that act against HCV on the molecular level.

Stochastic Runge–Kutta Rosenbrock type methods for SDE systems

In this paper we introduce a family of stochastic Runge–Kutta Rosenbrock (SRKR) type methods for multi-dimensional Itô stochastic differential equations (SDEs). The presented class of semi-implicit methods need less computational effort in comparison with some implicit ones. General order conditions for the coefficients and the random variables of the SRKR methods are obtained. Then a set of order conditions for a subclass of stochastic weak second order is given. Numerical examples are presented to demonstrate the efficiency and accuracy of the new schemes.

Model hierarchy-based optimal control of radiative heat transfer

We present a model hierarchy multilevel optimisation approach to solve an optimal boundary control problem in glass manufacturing. The process is modelled by radiative heat transfer and formulated as an optimal control problem restricted by partial differential algebraic equations (PDAEs) and additional control constraints. We consider a sequence of model approximations given by space-time dependent non-linear PDAEs of ascending accuracy. The different models allow for a model hierarchy-based optimisation approach, where the models are shifted automatically as the optimisation proceeds. We present a realisation of a multilevel generalised SQP method within the fully space-time adaptive optimisation environment Kardos using linearly implicit methods of Rosenbrock type and multilevel finite elements. We apply the optimal control algorithm to a glass cooling problem and present numerical experiments for the model hierarchy-based approach in two spatial dimensions and for a fully space-time adaptive optimisation in three spatial dimensions.

Rosenbrock-type methods with Inexact AMF for the time integration of advection–diffusion–reaction PDEs

The numerical integration of PDEs of Advection–Diffusion–Reaction type in several spatial variables in the MoL framework is considered. The spatial discretization is based on Finite Differences and the time integration is carried out by using splitting techniques applied to Rosenbrock-type methods. The focus here is to provide a way of making some refinements to the usual Approximate Matrix Factorization (AMF) when it is applied to some well-known Rosenbrock-type methods. The proposed AMF-refinements provide new methods in a natural way and some of these methods belong to the class of the W-methods. Interesting stability properties of the resulting methods are proved and a few numerical experiments on some important non-linear PDE problems with applications in Physics are carried out.

High-order quasi-static finite element computations in space and time with application to finite strain viscoelasticity

On the one hand, high-order quasi-static finite elements have been developed on the basis of isogeometric analysis or hierarchical shape functions based on integrated Legendre-polynomials, which display favorable behavior in comparison to linear element formulations. If we consider constitutive models of evolutionary-type such as elastoplasticity, viscoplasticity or viscoelasticity, only first order methods are employed to integrate the ordinary-differential equations at the Gauss-point level, leading to errors caused by the time-integration. On the other hand, high-order time integration methods have been devised to treat the resulting system of differential–algebraic equations (DAE-system) following the spatial discretization. This, however, is only done using spatial discretizations with linear or quadratic h-elements based on Lagrange polynomials or related mixed element formulations. In this article both approaches are combined using a p-version finite element approach on the basis of hierarchical shape functions and the resulting DAE-system is solved by means of high-order one-step methods. The first step in this treatise is to apply stiffly accurate, diagonally-implicit Runge–Kutta methods combined with the Multilevel-Newton algorithm. Since the computational outlay for computing the tangential stiffness matrix is very high in the case of p-version finite elements, Rosenbrock-type methods are applied as well, which lead to a completely iteration-free technique. We then compare the two approaches. The next step is to investigate three-dimensional examples showing the properties on the basis of a constitutive model in finite strain viscoelasticity.

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.

Performance assessment of exponential Rosenbrock methods for large systems of ODEs

This article studies time integration methods for stiff systems of ordinary differential equations of large dimension. For such problems, implicit methods generally outperform explicit methods because the step size is usually less restricted by stability constraints. Recently, however, a family of explicit methods, called exponential integrators, have become popular for large stiff problems due to their favourable stability properties and the rapid convergence of non-preconditioned Krylov subspace methods for computing matrix-vector products involving exponential-like functions of the Jacobian matrix. In this article, we implement the so-called exponential Rosenbrock methods using Krylov subspaces. Numerical experiments on a challenging real-world test problem reveal that these methods are a promising preconditioner-free alternative to well-established approaches based on preconditioned Newton--Krylov implementations of the backward differentiation formulas. References E. J. Carr, T. J. Moroney and I. W. Turner. Efficient simulation of unsaturated flow using exponential time integration. Appl. Math. Comput., 217(14): 6587--6596, 2011. doi:10.1016/j.amc.2011.01.041 E. J. Carr, I. W. Turner and M. Ilic. Krylov subspace approximations for the exponential Euler method: error estimates and the harmonic Ritz approximant. ANZIAM Journal, 52: C612--C627, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3938 E. J. Carr, I. W. Turner and P. Perre. A variable step size Jacobian-free exponential integrator for simulating transport in heterogeneous porous media: application to wood drying. J. Comput. Phys., 233: 66-82, 2013. doi:10.1016/j.jcp.2012.07.024 Sundials user documentation. https://computation.llnl.gov/casc/sundials/documentation/documentation.html M. Hochbruck, C. Lubich and H. Selhofer. Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput., 19(5): 1552--1574, 1998. doi:10.1137/S1064827595295337 M. Hochbruck, A. Ostermann and J. Schweitzer. Exponential Rosenbrock-type methods. SIAM J. Numer. Anal., 47(1): 786--803, 2009. doi:10.1137/080717717 C. T. Kelley. Solving nonlinear equations with Newton's method. SIAM, USA, 2003. D. A. Knoll and D. E. Keyes. Jacobian-free Newton--Krylov methods: a survey of approaches and applications. J. Comput. Phys., 193(2): 357--397, 2004. doi:10.1016/j.jcp.2003.08.010 M. Tokman. Efficient integration of large stiff systems of ODEs with exponential propagation iteration (EPI) methods. J. Comput. Phys., 213: 748--776, 2006. doi:10.1016/j.jcp.2005.08.032

Efficient simulation of geothermal processes in heterogeneous porous media based on the exponential Rosenbrock–Euler and Rosenbrock-type methods

Simulation of geothermal systems is challenging due to coupled physical processes in highly heterogeneous media. Combining the exponential Rosenbrock–Euler method and Rosenbrock-type methods with control-volume (two-point flux approximation) space discretizations leads to efficient numerical techniques for simulating geothermal systems. In terms of efficiency and accuracy, the exponential Rosenbrock–Euler time integrator has advantages over standard time-discretization schemes, which suffer from time-step restrictions or excessive numerical diffusion when advection processes are dominating. Based on linearization of the equation at each time step, we make use of matrix exponentials of the Jacobian from the spatial discretization, which provide the exact solution in time for the linearized equations. This is at the expense of computing the matrix exponentials of the stiff Jacobian matrix, together with propagating a linearized system. However, using a Krylov subspace or Léja points techniques make these computations efficient.The Rosenbrock-type methods use the appropriate rational functions of the Jacobian of the ODEs resulting from the spatial discretization. The parameters in these schemes are found in consistency with the required order of convergence in time. As a result, these schemes are A-stable and only a few linear systems are solved at each time step. The efficiency of the methods compared to standard time-discretization techniques are demonstrated in numerical examples.

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.

Adaptive space and time numerical simulation of reaction–diffusion models for intracellular calcium dynamics

Adaptivity in space and time for the numerical simulation of stochastic and deterministic equations for intracellular calcium dynamics is presented. The modeling of diffusion, reaction and membrane transport of calcium ions in cells leads to a system of reaction–diffusion equations. We describe the modulation of cytosolic and ER calcium concentrations close to the membrane of the cell organelle.A conforming piecewise linear finite element method is used for the spatial discretization. Linearly implicit methods of Rosenbrock type are used for the time integration. We adopt a hybrid algorithm to solve the stochastic part. The space grid is adjusted to the strong localization of the calcium release following stochastic channel transitions. By automatically adapting the spatial meshes and time steps to the proper scales during the transition of channel states, the method accurately resolves the evolution of intracellular calcium concentrations as well as buffer concentrations. This article emphasizes adaptive and efficient hybrid numerical simulations in two space dimensions. The presented work establishes the basis for future simulations in a realistic 3D geometry.

Comparison of diagonal-implicit, linear-implicit and half-explicit Runge–Kutta methods in non-linear finite element analyses

The method of vertical lines in the case of quasi-static solid mechanics applying constitutive models of evolutionary-type yields after the spatial discretization by means of finite elements a system of differential-algebraic equations. It is of substantial interest how fast, accurate, and stable such computations can be carried out. Moreover, the questions are how simple the implementation can be done and how susceptible a procedure is to programming errors. In this article, this is investigated for half-explicit Runge–Kutta methods that are applied to small and finite strain viscoelasticity. The advantage of the method is given by a non-iterative scheme on element level. Additionally, it turns out that for models where linear elasticity is one ingredient in the constitutive model, the method leads to only one required LU-decomposition at the beginning of the entire computation, and in each time step, only one back-substitution step has to be carried out. This outperforms current finite element computations. Order investigations of various integration schemes and the automatic step-size behavior are studied. This new proposal is compared to a classical Backward-Euler implementation, high-order stiffly accurate diagonally implicit Runge–Kutta, and recently proposed Rosenbrock-type methods. Advantages and disadvantages of the applied schemes are discussed.

A comparison of time adaptive integration methods for small and large strain viscoelasticity

AbstractIn quasistatic solid mechanics the initial boundary value problem has to be solved in the space and time domain. The spatial discretization is done using finite elements. For the temporal discretization three different classes of Runge‐Kutta methods are compared. These methods are diagonally implicit Runge‐Kutta schemes (DIRK), linear implicit Runge‐Kutta methods (Rosenbrock type methods) and half‐explicit Runge‐Kutta schemes (HERK). It will be shown that the application of half‐explicit or linear‐implicit Runge‐Kutta methods can enormously reduce the computational time in particular situations. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Optimal control of radiative heat transfer in glass cooling with restrictions on the temperature gradient

SUMMARYThis paper is motivated by an optimal boundary control problem for the cooling process of molten and already formed glass down to room temperature. The high temperatures at which glass is processed demand to include radiative heat transfer in the computational model. Since the complete radiative heat transfer equations are too complex for optimization purposes, we use simplified approximations of spherical harmonics coupled with a practically relevant frequency bands model. The optimal control problem is considered as a partial differential algebraic equation (PDAE)‐constrained optimization problem with box constraints on the control. In this paper, we augment the objective by a functional depending on the state gradient, which forces a minimization of thermal stress inside the glass. To guarantee consistent and grid‐independent values of the reduced objective gradient at the end of the cooling process, we pursue two approaches. The first includes the temperature gradient with a time‐dependent linearly decreasing weight. In the second approach, we augment the objective functional by the final state tracking and final state gradient optimization. To determine an optimal boundary control, we apply a projected gradient method with the Armijo step size rule. The reduced objective gradient is computed by the continuous adjoint approach. The arising time‐dependent PDAEs are numerically solved by variable step size one‐step methods of Rosenbrock type in time and adaptive multilevel finite elements in space. We present two‐dimensional numerical results for an infinitely long glass block and compare the two different approaches derived to ensure consistency at the end of the cooling process. Copyright © 2011 John Wiley & Sons, Ltd.

A Class of Linearly Implicit Numerical Methods for Solving Stiff Ordinary Differential Equations~!2009-10-30~!2010-02-02~!2010-04-06~!

We introduce ABC-schemes, a new class of linearly implicit one-step methods for numerical integration of stiff ordinary differential equation systems. Formulas of ABC-schemes invoke the Jacobian of differential system similary to the methods of Rosenbrock type, but unlike the latter they include also the square of the Jacobian matrix.