Explicit Singly Diagonally Implicit Runge-Kutta Research Articles

A comparative study of sensitivity computations in ESDIRK-based optimal control problems

This paper compares the impact of iterated and direct approaches to sensitivity computation in fixed step-size explicit singly diagonally implicit Runge–Kutta (ESDIRK) methods when applied to optimal control problems (OCPs). We strictly use the principle of internal numerical differentiation (IND) for the iterated approach, i.e., reusing iteration matrix factorizations, the number of Newton-type iterations, and Newton iterates, to compute the sensitivities. The direct method computes the sensitivities without using the Newton schemes. We compare the impact of these sensitivity computations in OCPs for the quadruple tank system (QTS). We discretize the OCPs using multiple shooting and solve these with a sequential quadratic programming (SQP) solver. We benchmark the iterated and direct approaches against a base case. This base case applies the ESDIRK methods with exact Newton schemes and a direct approach for sensitivity computations. In these OCPs, we vary the number of integration steps between control intervals and evaluate the performance based on the number of SQP and QPs iterations, KKT violations, function evaluations, Jacobian updates, and iteration matrix factorizations. We also provide examples using the continuous-stirred tank reactor (CSTR), and the IPOPT algorithm instead of the SQP. For OCPs solved using SQP, the QTS results show the direct method converges only once, while the iterated approach and base case converges in all situations. Similar results are seen with the CSTR. Using IPOPT, both the iterated approach and base case converge in all cases. In contrast, the direct method only converges in all cases regarding the CSTR.

Numerical Methods for Optimal Boundary Control of Advection-Diffusion-Reaction Systems

This paper considers the optimal boundary control of chemical systems described by advection-diffusion-reaction (ADR) equations. We use a discontinuous Galerkin finite element method (DG-FEM) for the spatial discretization of the governing partial differential equations, and the optimal control problem is directly discretized using multiple shooting. The temporal discretization and the corresponding sensitivity calculations are achieved by an explicit singly diagonally-implicit Runge Kutta (ESDIRK) method. ADR systems arise in process systems engineering and their operation can potentially be improved by nonlinear model predictive control (NMPC). We demonstrate a numerical approach for the solution to their optimal control problems (OCPs) in a chromatography case study. Preparative liquid chromatography is an important downstream process in biopharmaceutical manufacturing. We show that multi-step elution trajectories for batch processes can be optimized for economic objectives, providing superior performance compared to classical gradient elution trajectories.

A flux reconstruction kinetic scheme for the Boltzmann equation

It is challenging to solve the Boltzmann equation accurately due to the extremely high dimensionality and nonlinearity. This paper addresses the idea and implementation of the first flux reconstruction method for high-order Boltzmann solutions. Based on the Lagrange interpolation and reconstruction, the kinetic upwind flux functions are solved simultaneously within physical and particle velocity space. The fast spectral method is incorporated to solve the full Boltzmann collision integral with a general collision kernel. The explicit singly diagonally implicit Runge-Kutta (ESDIRK) method is employed as time integrator and the stiffness of the collision term is smoothly overcome. Besides, we ensure the shock capturing property by introducing a self-adaptive artificial dissipation, which is derived naturally from the effective cell Knudsen number at the kinetic scale. As a result, the current flux reconstruction kinetic scheme can be universally applied in all flow regimes. Numerical experiments including wave propagation, normal shock structure, one-dimensional Riemann problem, Couette flow and lid-driven cavity will be presented to validate the scheme. The order of convergence of the current scheme is clearly identified. The capability for simulating cross-scale and non-equilibrium flow dynamics is demonstrated.

Higher-order additive Runge–Kutta schemes for ordinary differential equations

Two new implicit–explicit, additive Runge–Kutta (ARK2) methods are given with fourth- and fifth-order formal accuracies, respectively. Both combine explicit Runge–Kutta (ERK) methods with explicit, singly-diagonally implicit Runge–Kutta (ESDIRK) methods and include an embedded method for error control. The two methods have ESDIRKs which are internally L-stable on stages three and higher, have only modestly negative eigenvalues to the stage and step algebraic-stability matrices and have stage-order two. To improve computational efficiency, the fourth-order method has a diagonal coefficient of 0.1235. This is done to offset much of the extra computational cost of an extra stage by facilitating iterative convergence at each stage. Linear stability domains for both ERK methods have been made quite large and the dominant coupling stability term between the stability of the ESDIRK and ERK for very stiff modes has been removed. Though the fourth-order method is one of the best all-around fourth-order IMEX ARK2 of which we are aware, the fifth-order method is likely best suited to mildly stiff problems with tight error tolerances. Methods are tested using the Van der Pol and Kaps' singular-perturbation problems. Results suggest that these new methods represent an improvement over existing methods of the same class.

A high-order Discontinuous Galerkin solver for unsteady incompressible turbulent flows

In this work we investigate the use of adaptive linearly implicit Rosenbrock-type Runge–Kutta and Explicit Singly Diagonally Implicit Runge–Kutta schemes to integrate in time high-order Discontinuous Galerkin space discretizations of the incompressible Navier–Stokes (INS) and Reynolds Averaged Navier–Stokes (URANS) equations. The objective of this activity is to assess the efficiency and accuracy of the considered schemes coupled with a time-step adaptation technique for incompressible URANS simulations. The schemes have been first investigated for the computation of the laminar travelling waves and of the turbulent flow around a circular cylinder at a Reynolds number Re=5×104, verifying the convergence order, a simple relation to set the system tolerance starting from the tolerance of the adaptation strategy, and their computational efficiency. Finally, the best scheme resulting from our analysis has been applied to the URANS simulation of the flow through a vertical axis wind turbine, comparing the results with CFD and experimental data available in literature.

Oil Reservoir Production Optimization using Single Shooting and ESDIRK Methods

Conventional recovery techniques enable recovery of 10-50% of the oil in an oil field. Advances in smart well technology and enhanced oil recovery techniques enable significant larger recovery. To realize this potential, feedback model-based optimal control technologies are needed to manipulate the injections and oil production such that flow is uniform in a given geological structure. Even in the case of conventional water flooding, feedback based optimal control technologies may enable higher oil recovery than with conventional operational strategies. The optimal control problems that must be solved are large-scale problems and require specialized numerical algorithms. In this paper, we combine a single shooting optimization algorithm based on sequential quadratic programming (SQP) with explicit singly diagonally implicit Runge-Kutta (ESDIRK) integration methods and a continuous adjoint method for sensitivity computation. We demonstrate the procedure on a water flooding example with conventional injectors and producers.

Adaptive Stepsize Control in Implicit Runge-Kutta Methods for Reservoir Simulation

This paper concerns predictive stepsize control applied to high order methods for temporal discretization in reservoir simulation. The family of Runge-Kutta methods is presented and in particular the explicit singly diagonally implicit Runge-Kutta (ESDIRK) methods are described. A predictive stepsize adjustment rule based on error estimates and convergence control of the integrated iterative solver is presented. We try to improve the predictive stepsize control by smoothing the stepsize sequence through combining the control of error with the control of convergence.

A functionally fitted three-stage explicit singly diagonally implicit Runge-Kutta method

A special class of Runge-Kutta(-Nystrom) methods called functionally fitted (or functional fitting) Runge-Kutta (FRK) methods has recently been proposed by the author. This class of methods is designed to be exact, if the solution of the equation to be solved is an element of the linear space of given functions, which is called the basis functions. The purpose of this article is to develop a functionally fitted Runge-Kutta method that is cheap to implement. The method proposed in this paper is a three-stage explicit singly diagonally implicit Runge-Kutta (ESDIRK) method, which requires oneLU decomposition per step. This method is exact if the basis functions are properly chosen, and is moderately accurate even if the choice of the functions is inappropriate, since the method is shown to be of order 4 for general cases. An embedded pair of this type is also developed. Several numerical experiments show the superiority of the methods to conventional ones for the particular case that a suitable set of the basis functions can be found.

Implicit Time Integration Schemes for the Unsteady Compressible Navier–Stokes Equations: Laminar Flow

The accuracy and efficiency of several lower and higher order time integration schemes are investigated for engineering solution of the discretized unsteady compressible Navier–Stokes equations. Fully implicit methods tested are either the backward differentiation formulas (BDF) or stage-order two, explicit, singly diagonally implicit Runge–Kutta (ESDIRK) methods. For this comparison an unsteady two-dimensional laminar flow problem is chosen: flow around a circular cylinder at Re=1200. At temporal error tolerances consistent with engineering simulation, ϵ≈10 −1–10 −2, first-order implicit Euler (BDF1) is uncompetitive. While BDF3 is quite efficient, its lack of A-stability may be problematic in the presence of convection. At these same error tolerances, the fourth-order ESDIRK scheme is 2.5 times more efficient than BDF2. It is concluded that reliable integration is most efficiently provided by fourth-order Runge–Kutta methods for this problem where order reduction is not observed. Efficiency gains are more dramatic at smaller tolerances.

Additive Runge–Kutta schemes for convection–diffusion–reaction equations

Additive Runge–Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection–diffusion–reaction (CDR) equations. Accuracy, stability, conservation, and dense-output are first considered for the general case when N different Runge–Kutta methods are grouped into a single composite method. Then, implicit–explicit, ( N=2), additive Runge–Kutta (ARK 2) methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge–Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge–Kutta method (ERK). Coupling error terms of the partitioned method are of equal order to those of the elemental methods. Derived ARK 2 methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z [ I] →−∞, and retain high stability efficiency in the absence of stiffness, z [ I] →0. Extrapolation-type stage-value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK 2 error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK 2 methods.

A four-stage index 2 Diagonally Implicit Runge–Kutta method

High index Differential Algebraic Equations (DAEs) force standard numerical methods to lower order. Implicit Runge–Kutta methods such as RADAU5 handle high index problems but their fully implicit structure creates significant overhead costs for large problems. Singly Diagonally Implicit Runge–Kutta (SDIRK) methods offer lower costs for integration. This paper derives a four-stage, index 2 Explicit Singly Diagonally Implicit Runge–Kutta (ESDIRK) method. By introducing an explicit first stage, the method achieves second order stage calculations. After deriving and solving appropriate order conditions, numerical examples are used to test the proposed method using fixed and variable step size implementations.