Semi-implicit Runge-Kutta Research Articles

Hp-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows: Part I. Multilevel analysis

The hp-Multigrid as Smoother algorithm (hp-MGS) for the solution of higher order accurate space–time discontinuous Galerkin discretizations of advection dominated flows is presented. This algorithm combines p-multigrid with h-multigrid at all p-levels, where the h-multigrid acts as smoother in the p-multigrid. The performance of the hp-MGS algorithm is further improved using semi-coarsening in combination with a new semi-implicit Runge–Kutta method as smoother. A detailed multilevel analysis of the hp-MGS algorithm is presented to obtain more insight into the theoretical performance of the algorithm. As model problem a fourth order accurate space–time discontinuous Galerkin discretization of the advection–diffusion equation is considered. The multilevel analysis shows that the hp-MGS algorithm has excellent convergence rates, both for steady state and time-dependent problems, and low and high cell Reynolds numbers, including highly stretched meshes.

HP-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II: Optimization of the Runge–Kutta smoother

Using a detailed multilevel analysis of the complete hp-Multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multigrid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit Runge–Kutta smoother, such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints. The Runge–Kutta coefficients for a wide range of cell Reynolds numbers and a detailed analysis of the performance of the hp-MGS algorithm are presented. In addition, the computational complexity of the hp-MGS algorithm is investigated. The hp-MGS algorithm is tested on a fourth order accurate space–time discontinuous Galerkin finite element discretization of the advection–diffusion equation for a number of model problems, which include thin boundary layers and highly stretched meshes, and a non-constant advection velocity. For all test cases excellent multigrid convergence is obtained.

A hybrid method of semi-Lagrangian and additive semi-implicit Runge–Kutta schemes for gyrokinetic Vlasov simulations

A hybrid method of semi-Lagrangian and additive semi-implicit Runge–Kutta schemes is developed for gyrokinetic Vlasov simulations in a flux tube geometry. The time-integration scheme is free from the Courant–Friedrichs–Lewy condition for the linear advection terms in the gyrokinetic equation. The new method is applied to simulations of the ion-temperature-gradient instability in fusion plasmas confined by helical magnetic fields, where the parallel advection term severely restricts the time step size for explicit Eulerian schemes. Linear and nonlinear results show good agreements with those obtained by using the explicit Runge–Kutta–Gill scheme, while the new method substantially reduces the computational cost.

Semi-Implicit Runge–Kutta Schemes: Development and Application to Compressible Channel Flow

Two semi-implicit six-stage Runge–Kutta algorithms are developed for the simulation of wall-bounded flows. Using these schemes, time integration is implicit in the wall-normal direction, and explicit in the other directions, to relax the time step constraint due to the fine mesh near the wall. The explicit subscheme is a six-stage fourth-order low-storage Runge–Kutta scheme. Based on analysis in Fourier space and results obtained for propagation test cases, the semi-implicit schemes are shown to be of order 3 and, for waves discretized by a number of points per period between 4 and 16, to be as accurate as, or more accurate than, the standard explicit fourth-order Runge–Kutta algorithm in terms of dissipation and dispersion. The large-eddy simulation of a compressible turbulent channel flow at a friction Reynolds number of 360 and a Mach number of 0.1 is then carried out with one of the proposed algorithms. The computational time is reduced by a factor 1.33 with respect to a large-eddy simulation using the explicit subscheme in all directions. Wall-pressure and velocity spectra from the large-eddy simulation are presented to give insights into the flow turbulent structures. In particular, wave number–frequency spectra are calculated. Acoustic components appear to be identified in these spectra.

A linearly semi-implicit compact scheme for the Burgers–Huxley equation

In this paper, a linearly semi-implicit compact scheme is developed for the Burgers–Huxley equation. The equation is decomposed into two subproblems, i.e. a Burgers equation and a nonlinear ODE, by the operator splitting technique. The Burgers equation is solved by a linearly self-starting compact scheme which is fourth-order accurate in space and second-order accurate in time. The nonlinear ODE is discretized by a third-order semi-implicit Runge–Kutta method, which possesses good numerical stability with low computational cost. The numerical experiments show that the scheme provides the expected convergence order. Finally, several experiments are conducted to simulate the solutions of the Burgers–Huxley equation to validate our numerical method.

The Reduced Basis Technique as a Coarse Solver for Parareal in Time Simulations

In this paper, we extend the reduced basis methods for parameter dependent problems to the parareal in time algorithm introduced by Lions et al. [12] and solve a nonlinear evolutionary parabolic partial difierential equation. The flne solver is based on the flnite element method or spectral element method in space and a semi-implicit Runge-Kutta scheme in time. The coarse solver is based on a semi-implicit scheme in time and the reduced basis approximation in space. O†ine-online procedures are developed, and it is proved that the computational complexity of the on-line stage depends only on the dimension of the reduced basis space (typically small). Parareal in time algorithms based on a multi-grids flnite element method and a multi-degrees flnite element method are also presented. Some numerical results are reported.

Design and Implementation of Predictors for Additive Semi-Implicit Runge–Kutta Methods

Space discretization of some time-dependent partial differential equations gives rise to stiff systems of ordinary differential equations. In this case, implicit methods should be used, and therefore, in general, nonlinear systems must be solved. The solutions to these systems are approximated by iterative schemes, and, in order to obtain an efficient code, good initializers should be used. Recently, a parallel code based on some Runge-Kutta and additive Runge-Kutta methods has been constructed, focusing especially on additive semi-implicit Runge-Kutta (ASIRK) methods. The aim of the present paper is to develop efficient initializers for these methods.

Modeling of fixed bed adsorption of phenols on granular activated carbon

Mathematical modeling of liquid phase adsorption of phenols in fixed beds of granular activated carbon was investigated. The model considered the effects of axial diffusion in the fluid, the external film and internal diffusional mass transfer resistances of the particles, and the nonlinear adsorption isotherm of Freundlich. It was shown that the analysis of a complex multicomponent adsorption system could be simplified by converting it into a pseudo single-component adsorption system. This was achieved by lumping the concentrations of the components together as one single parameter, chemical oxygen demand. The resulting model equations were solved using the orthogonal collocation method and third-order semi-implicit Runge-Kutta method combined with a step-size adjustment strategy. Excellent agreement between simulated results and pilot plant data was obtained. Also, the breakthrough profiles revealed the formation of a primary monomolecular layer on the adsorbent surface.

Numerical simulation of turbulence transition and sound radiation for flow through a rigid glottal model

Large eddy simulation (LES)-based computational aeroacoustics techniques were applied to a static model of the human glottis, idealized here as a planar channel with an orifice, to study flow-acoustic interactions related to speech. Rigid models of both converging and diverging glottal passages, each featuring a 20 deg included angle and a minimal glottal diameter of 0.04 cm, with an imposed transglottal pressure of 15 cm H2O, were studied. The Favre-filtered compressible Navier-Stokes equations were integrated for this low-Mach-number flow using an additive semi-implicit Runge-Kutta method and a high-order compact finite-difference scheme with characteristic-based nonreflecting boundary conditions and a multiblock approach. Flow asymmetries related to the Coanda effect and transition to turbulence, as well as the far-field sound, were captured. Acoustic-analogy-based far-field sound predictions were compared with direct simulations and showed that dipole sources, arising from unsteady flow forces exerted on the glottal walls, are primarily responsible for the tonal sound observed in the divergent glottis case.

A Semi-Implicit Runge–Kutta Time-Difference Scheme for the Two-Dimensional Shallow-Water Equations

Abstract A semi-implicit, two time-level, three-step iterative time-difference scheme is proposed for the two-dimensional nonlinear shallow-water equations in a conservative flux form. After a semi-implicit linearization of the governing equations, the linear gravity wave terms are time discretized implicitly using a second-order trapezoidal scheme applied over each iterative step, whereas the nonlinear terms including horizontal advection and other terms left over from the semi-implicit linearization are time discretized explicitly using a third-order Runge–Kutta scheme. The effectiveness of the scheme in terms of numerical accuracy, stability, and efficiency is established through a forced initial-boundary value problem studied using a two-dimensional shallow-water model.

Direct numerical simulation of the turbulent flow in an elliptical pipe

The turbulent flow in a pipe with an elliptical cross section is directly simulated at Re = 4000 (where the Reynolds number Re is calculated in terms of the mean velocity and the hydraulic diameter). The incompressible Navier-Stokes equations are solved in curvilinear orthogonal coordinates by using a central-difference approximation in space and a third-order accurate semi-implicit Runge-Kutta method for time integration. The discrete equations inherit some properties of the original differential equations, in particular, the neutrality of the convective terms and of the pressure gradient in the kinetic energy production. The distributions of the mean and fluctuation characteristics of the turbulent motion over the pipe’s cross section are computed.

An additive semi-implicit Runge–Kutta family of schemes for nonstiff systems

An additive semi-implicit third-order Runge–Kutta family of schemes for nonstiff systems of equations is presented. These schemes involve a single implicit stage and two explicit stages and posses a linear stability domain that is larger than that of explicit Runge–Kutta schemes of the same order. The improved stability domain can be useful in some problems involving large systems of equations where a few components impose too stringent time-step limitations.

Finite-difference method for incompressible Navier–Stokes equations in arbitrary orthogonal curvilinear coordinates

A finite-difference method for solving three-dimensional time-dependent incompressible Navier–Stokes equations in arbitrary curvilinear orthogonal coordinates is presented. The method is oriented on turbulent flow simulations and consists of a second-order central difference approximation in space and a third-order semi-implicit Runge–Kutta scheme for time advancement. Spatial discretization retains some important properties of the Navier–Stokes equations, including energy conservation by the nonlinear and pressure-gradient terms. Numerical tests cover Cartesian, cylindrical-polar, spherical, cylindrical elliptic and cylindrical bipolar coordinate systems. Both laminar and turbulent flows are considered demonstrating reasonable accuracy and stability of the method.

Direct numerical simulation of turbulent flows in eccentric pipes

A numerical algorithm was developed for solving the incompressible Navier-Stokes equations in curvilinear orthogonal coordinates. The algorithm is based on a central-difference discretization in space and on a third-order accurate semi-implicit Runge-Kutta scheme for time integration. The discrete equations inherit some properties of the original differential equations, in particular, the neutrality of the convective terms and the pressure gradient in the kinetic energy production. The method was applied to the direct numerical simulation of turbulent flows between two eccentric cylinders. Numerical computations were performed at Re = 4000 (where the Reynolds number Re was defined in terms of the mean velocity and the hydraulic diameter). It was found that two types of flow develop depending on the geometric parameters. In the flow of one type, turbulent fluctuations were observed over the entire cross section of the pipe, including the narrowest gap, where the local Reynolds number was only about 500. The flow of the other type was divided into turbulent and laminar regions (in the wide and narrow parts of the gap, respectively).

Compressible large eddy simulations of wall-bounded turbulent flows using a semi-implicit numerical scheme for low Mach number aeroacoustics

Large eddy simulations (LES) of low-speed, wall-bounded turbulent flows were conducted by numerically integrating the compressible Navier–Stokes equations in a generalized curvilinear coordinate system. An efficient numerical scheme based on a third-order additive semi-implicit Runge–Kutta method for time advancement and a sixth-order accurate, compact finite-difference scheme for spatial discretization were used. The convective terms in the wall-normal direction were treated implicitly to remove the time-step limitation associated with the use of fine meshes in the near-wall region for high Reynolds number viscous flows. The dynamic Smagorinsky subgrid-scale eddy viscosity model was used to close the filtered equations. Generalized characteristic-based non-reflecting boundary conditions were used together with grid stretching and enhanced damping in the exit zone. The accuracy and efficiency of the numerical scheme was assessed by simple acoustic model problems and by comparing LES predictions for fully developed turbulent channel flow and turbulent separated flow in an asymmetric diffuser to previous direct numerical simulation (DNS) and experimental data, respectively. LES predictions for both flows were in reasonable agreement with the DNS and experimental mean velocity and turbulence statistics. The findings suggest that the numerical approach employed here offers comparable accuracy to similar recent studies at approximately one-third of the computational cost and may provide both an accurate and efficient way to conduct computational aeroacoustics studies for low Mach number, confined turbulent flows.

Periodically operated trickle-bed reactor for EAQs hydrogenation: Experiments and modeling

The influence of periodic operation on a consecutive reaction, the hydrogenation of 2-ethylanthraquinones (EAQs) over Pd / Al 2 O 3 , on a laboratory-scale trickle-bed reactor (TBR) was studied. The effects of operating parameters including cycle period, split, pressure, temperature, and time-average flow rate on the performance were experimentally examined in comparison with the steady-state operation. The results showed that under the interested operating conditions the conversion and the selectivity improved by 3–21% and 1–12%, respectively. A dynamic model consisting of a set of partial differential equations (PDEs) was developed to simulate the periodic operation of TBR for EAQs hydrogenation. The PDEs were converted into a set of ordinary differential equations (ODEs) using the method of lines (MOL) and then numerically solved by the semi-implicit Runge–Kutta method. The developed model was verified by simulating the effect of cycle period and split on the conversion and the selectivity enhancement and compared with the experimental results. It was found that the model was reliable and satisfactory when the cycle period was less than 200 s.

Numerical Solution of Binary Liquid-Phase Adsorption onto Porocel Clay Using Linear, Freundlich and Langmuir Isotherms

A generalized mathematical model is presented to describe the process of multi-component adsorption onto porous media in fixed beds. The model was applied to the binary adsorption, without reaction, of aromatics and sulphur compounds onto a fixed bed of Porocel clay for kerosene deodorization using linear, Freundlich and Langmuir isotherms independently. A computational scheme for the solution of the model equations is presented. The scheme is based on orthogonal collocation for spatial discretization of the resulting set of coupled hyperbolic and parabolic partial differential equations for the macro-and micro-system, respectively. Michelsen's modified third-order semi-implicit Runge-Kutta method combined with step-size adjustment strategy was used to integrate the resulting 4N ordinary differential equations. Excellent agreement between the simulated results and pilot plant data was obtained for the breakthrough profiles for the non-linear adsorption isotherms of Freundlich and Langmuir. No agreement was obtained for the linear isotherm. Also, using the Freundlich and Langmuir isotherms, the exit concentration of the less preferentially adsorbed component (aromatics) exceeded its inlet concentration to the adsorption column for a certain period. This is indicative of the behaviour of competitive multi-component adsorption: relative to aromatics, sulphur compounds are selectively adsorbed onto Porocel clay. The relationship between solid- and liquid-phase concentration profiles for the Freundlich isotherm revealed the formation of multiple adsorption layers upon the primary mono-molecular layer. Again, for the Freundlich isotherm, the structure of the profiles exhibited a highly pronounced maximum for sulphur. An experimental breakthrough time of 8 h was also predicted for both aromatics and sulphur compounds using the non-linear Freundlich and Langmuir isotherms.

SENSITIVITY ANALYSIS IN INDEX-1 DIFFERENTIAL ALGEBRAIC EQUATIONS BY ESDIRK METHODS

Dynamic optimization by multiple shooting requires integration and sensitivity calculation. A new semi-implicit Runge-Kutta algorithm for numerical sensitivity calculation of index-1 DAE systems is presented. The algorithm calculates sensitivities with respect to problem parameters and initial conditions, exploiting the special structure of the sensitivity equations. The algorithm is a one-step method which makes it especially efficient compared to multiple-step methods when frequent discontinuities are present. These advantages render the new algorithm particularly suitable for dynamic optimization and nonlinear model predictive control. The algorithm is tested on the Dow Chemicals benchmark problem.

Modelling and Simulation of Ternary Liquid-Phase Adsorption onto Activated Carbon

A computational procedure is presented for solving the set of rigid hyperbolic and parabolic partial differential equations describing the simultaneous adsorption of a ternary system in a column packed with adsorbent particles using the non-linear adsorption isotherms of Fritz and Schluender. The model equations account for the effects of axial diffusion in the fluid and the film and internal diffusional mass-transfer resistances of the particles. Orthogonal collocation and Michelsen's modified third-order semi-implicit Runge-Kutta method combined with a step-size adjustment strategy were used to solve the general form of the resulting 6N coupled ordinary differential equations for the simultaneous adsorption of butan-2-ol, t-amyl alcohol and phenol in fixed beds. The simulated results obtained from this model were compared with experimental data. Excellent agreement between the simulated results and previously published experimental data was obtained. The breakthrough profiles were also indicative of the...

Efficient sensitivity computation for nonlinear model predictive control

Dynamic optimization by multiple shooting requires integration and sensitivity calculation of the model ODEs as well as the cost function integral. A new semi-implicit Runge-Kutta algorithm for numerical sensitivity calculation of stiff systems of ordinary differential equations (ODEs) is presented. The algorithm calculates sensitivities with respect to problem parameters and initial conditions, exploiting the special structure of the sensitivity equations. The new algorithm is compared to an extrapolation method and to the more established BDF based approaches. Several advantages of the new approach are demonstrated, especially when frequent discontinuities are present. These advantages renders the new algorithm particularly suitable for dynamic optimization and nonlinear model predictive control.