High-order Time Integration Schemes Research Articles

High order time discretisation methods for incompressible Navier–Stokes equations

AbstractIn this paper we solve incompressible Navier–Stokes equations with higher order time integration schemes such as Radau‐IIA and Lobatto‐IIIC methods. We compare the results with 4th order DIRK and ROW methods to show the advantages of higher order time integration methods. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Analytical and numerical treatment of electro-thermo-mechanical coupling

Single-field problems are well understood concerning their numerical treatment and their numerical problems. In coupled situations, this is not always the case. This article is an attempt to fill the gap for the three-field problem of small strain electro-thermo-elasticity. Apart from the quasi-static equilibrium conditions, also, the nonlinear heat equation and the stationary but temperature-dependent electrical current equation are treated. The following aspects are investigated: First, analytical equations for code verification purposes are provided for particular subproblems. Secondly, we apply the method of vertical lines, which leads to a system of differential-algebraic equations—after the spatial discretization—that can be solved using stiffly accurate, diagonally implicit Runge–Kutta methods. This enables to apply high-order time integration schemes and to make use of step-size control by embedded schemes. Here, the known problems of the connection between mesh- and the temporal step-size must be studied, also focusing on the order reduction caused by nonlinear Dirichlet boundary conditions. Additionally, the resulting system of nonlinear equations is treated with different methods such as the Newton–Raphson scheme, the Newton–Chord method and a Newton–Chord method with Aitken relaxation leading to very efficient computations.

Space-mapping in fluid–structure interaction problems

Transient response due to gust loads can lead to structural failure despite the fact that the aero-elastic system is asymptotically stable. Unsteady aeroelastic analysis should therefore be included in the load calculation cycle of the aircraft design process. Especially in the transonic regime, partitioned strong coupling algorithms perform better than loose coupling algorithms and allow larger time-steps in the unsteady simulation. However, the simulation of high fidelity unsteady fluid–structure interaction using strong coupling algorithms is currently too expensive in order to be useful in industry.To accelerate high fidelity partitioned fluid–structure interaction simulations we apply space-mapping, which is a technique that originates from the field of multi-fidelity optimization. Without loss of generality we assume the availability of a cheap low fidelity fluid solver and an expensive high fidelity fluid solver. The space-mapping approach is used to accelerate the high fidelity computation using black-box input/output information of both the high fidelity fluid solver and low fidelity fluid solver. In order to achieve this, a space-mapping function is defined on the fluid–structure interface which keeps track of the differences between the high fidelity model and the low fidelity model during the coupling iterations. Reformulating the root-finding problem on the fluid–structure interaction interface using the space-mapping function results in the Aggressive Space-Mapping algorithm.The Aggressive Space-Mapping algorithm is applied to 1-D and 2-D test cases in order to assess the speedup with respect to the Quasi-Newton Inverse Least Squares algorithm. The latter is considered to be the current state of the art in partitioned strong coupling algorithms. The observed speedup depends mainly on the type of FSI problem and the time step size. The maximum observed speedup is about 1.5.The application of the space-mapping technique to partitioned fluid–structure interaction problems is found to be a promising approach. The framework is non-intrusive and allows the reuse of existing solvers which is especially useful in an industrial environment. It is expected that the space-mapping technique can be combined with higher order time integration schemes that maintain accuracy over a large range of time step sizes.

A monotonic limiter and flux-corrected transport approach for high-order time integration schemes

A monotonic transport algorithm for a high-order time integration scheme is described in this paper. The algorithm is a modified version of an existing high-order time integration scheme, and is tested using a simple one-dimensional pulse and two-dimensional deformational flows. It is found that the new formulation can remove the error, caused by new maxima/minima and excessive smoothing, which occurs in scalar transport using the original extrapolation scheme. The results show that there may be potential for the high-order time integration scheme to be applied in numerical weather prediction models.

Numerical prediction of oil amount leaked from a damaged tank using two-dimensional moving particle simulation method

When an oil-spilling accident occurs at sea, it causes tremendous economic loss and environmental damages. To minimize these marine disasters, predicting the amount of oil-leaking is one of the most important things for the quick response and decision making in the early stage of the accident. In the present study, numerical investigation on the oil-leaking phenomena from a two-dimensional damaged tank was carried out to predict the amount of the oil leaking from a leakage hole of a rectangular tank by using the PNU-MPS(Pusan-National-University-modified Moving Particle Simulation) method (Lee et al., 2011). As a preliminary test to confirm the accuracy of the PNU-MPS method for the two-phase problem and to investigate the influence of the gradient model, numerical simulations for the Rayleigh–Taylor(R–T) instability were carried out By using an appropriate gradient model and applying a high-order time integration scheme, i.e. 4th-order Runge–Kutta scheme, it was found that the simulation results became closer to the experimental ones (Kim and Lee, 2001), by which the oil-leaking speed and Torricelli's factor relating the speed predicted by using the hydrostatic balance and the real leakage speed were measured and assessed.

Time-integration methods for finite element discretisations of the second-order Maxwell equation

This article deals with time integration for the second-order Maxwell equations with possibly non-zero conductivity in the context of the discontinuous Galerkin finite element method (DG-FEM) and the H(curl)-conforming FEM. For the spatial discretisation, hierarchic H(curl)-conforming basis functions are used up to polynomial order p=3 over tetrahedral meshes, meaning fourth-order convergence rate. A high-order polynomial basis often warrants the use of high-order time-integration schemes, but many well-known high-order schemes may suffer from a severe time-step stability restriction owing to the conductivity term. We investigate several possible time-integration methods from the point of view of accuracy, stability and computational work. We also carry out a numerical Fourier analysis to study the dispersion and dissipation properties of the semi-discrete DG-FEM scheme as well as the fully-discrete schemes with several of the time-integration methods. The dispersion and dissipation properties of the spatial discretisation and those of the time-integration methods are investigated separately, providing additional insight into the two discretisation steps.

A Discontinuous Galerkin Finite Element Time-Domain Method Modeling of Dispersive Media

A method is proposed for solving the time-dependent Maxwell's equations via the discontinuous Galerkin finite-element time-domain (DGFETD) method with dispersive media. An auxiliary differential equation (ADE) method is used to represent the constitutive relations. The method is applied to Drude materials, as well as to multiple pole Debye and Lorentz materials. An efficient implementation for high-order Runge-Kutta time integration schemes is presented. The method is validated and is shown to exhibit high-order convergence.

Performance of convolutional perfectly matched layers for pseudospectral time domain poroviscoelastic schemes

This paper presents an implementation of the convolutional perfectly matched layer for the velocity–stress formulation of poroviscoelastic equations in anisotropic media, with high-order time and space formulations. After its introduction by Bérenger in 1994 for Maxwell's equation, the perfectly matched layer (PML) quickly became the standard approach to absorb outgoing waves on a numerical mesh. Subsequent developments include generalization and formulation in terms of a convolutional operator and auxiliary differential equations that improve efficiency and allow for the implementation of high-order time integration schemes. In this work, a fourth-order Runge–Kutta scheme is employed. Also, with the convolutional formulation, the wavefields need not be split as in the original formalism. Such unsplit PMLs allow for the implementation with a pseudospectral operator as presented in this paper. Although efforts have been deployed for the optimization of PML parameters, such an approach is not always tractable, especially for seismic wave propagation because different types of waves interact with the medium. Hence, the performance of the implementation is evaluated through a series of numerical experiments. Tests are performed for waves impinging on the PML interface both at normal and grazing incidence, for both isotropic and anisotropic media. The results highlight the advantage of the convolutional formulation for waves at grazing incidence and show that anisotropic media require a larger number of absorbing layers to achieve performances comparable to those obtained in isotropic media.

High-order resolution Eulerian–Lagrangian simulations of particle dispersion in the accelerated flow behind a moving shock

This paper presents a computational study of the two-dimensional particle-laden flow developments of bronze particle clouds in the accelerated flow behind a moving normal shock. Particle clouds with a particle volume concentration of 4% are arranged initially in a rectangular, triangular and circular shape. Simulations are performed with a recently developed high-order resolution Eulerian–Lagrangian method that approximates the Euler equations governing the gas dynamics with the improved high order weighted essentially non-oscillatory (WENO-Z) scheme, while individual particles are traced in the Lagrangian frame using high-order time integration schemes. Reflected shocks form ahead of all the cloud shapes. The detached shock in front of the triangular cloud is weakest. At later times, the wake behind the cloud becomes unstable, and a two-dimensional vortex-dominated wake forms. Separated shear layers at the edges of the clouds pull particles initially out of the clouds that are consequently transported along the shear layers. Since flows separated trivially at sharp corners, particles are mostly transported out of the cloud into the flow at the sharp front corner of the rectangular cloud and at the trailing corner of the triangular cloud. Particles are transported smoothly out of the circular cloud, since it lacks sharp corners. At late times, the accelerated flow behind the running shock disperses the particles in cross-stream direction the most for the circular cloud, followed by the rectangular cloud and the triangular cloud.

High-Order Time-Integration Schemes with Explicit Time-Splitting Methods

Abstract New high-order time-integration schemes for fully elastic models are presented. The new schemes, formulated using the Richardson extrapolation that employs leapfrog-type schemes, can give a good performance for linear model problems and ensure overall stability when they are combined with a forward–backward scheme for fast waves. The new and existing schemes show differences in the order of accuracy. Thus, they can be useful for investigating the impacts of time-integration scheme accuracy on the performance of numerical models. The high-order schemes are found to play an important role in the improvement of high-resolution simulations, according to idealized tests. The new schemes are less efficient than other well-known schemes at moderate spatial resolutions. However, the new schemes can be more efficient than the existing schemes when the resolution becomes very high.

Eulerian–Lagrangian time-stepping methods for convection-dominated problems

The accuracy and efficiency of several lower and higher order time integration schemes in Eulerian–Lagrangian computations are investigated for solution of convection–diffusion problems with nonlinear reaction terms. The implementation of these schemes differs from their Eulerian counterparts in the fact that they are applied during each time step, along the characteristic curves rather than in the time direction. The major focus is to examine the computational characteristics of a class of implicit, explicit, and implicit–explicit time marching methods combined with the Eulerian–Lagrangian procedure. The obtained results for several benchmark problems are considered to be representative, and might be helpful for a fair rating of solution schemes, particularly in long time computations.

Higher-Order Time Integration Schemes for Aeroelastic Applications on Unstructured Meshes

Efficient techniques for computational aeroelasticity on unstructured meshes are investigated. These include the formulation of a fourth-order implicit Runge-Kutta time-integration scheme, a third-order backward-difference time-integration scheme, and a second-order backward-difference time-integration scheme for the coupled system of flow and structural equations. Strong coupling between the flow solution and structural equations is maintained by fully converging the fully coupled system at each time step or stage. The time-integration schemes are constructed such that discrete conservation is maintained in the flow solution in the presence of dynamically deforming meshes, by verifying the discrete geometric conservation law. Efficient multigrid solution techniques are devised for solving both the governing flow equations and the mesh motion equations, using the same agglomerated coarse levels for both problems. The flutter boundary prediction of the AGARD 445.6 wing is used to demonstrate the accuracy and efficiency of these techniques as well as the sensitivity of flutter boundary prediction to time-step size and temporal errors.

Implicit and Explicit Higher-Order Time Integration Schemes for Fluid-Structure Interaction Computations

In this paper higher order time integration schemes are applied to fluid-structure interaction (FSI) simulations. For a given accuracy, we investigate the efficiency of higher order time integration schemes compared to lower order methods. In the partitioned FSI simulations on a one-dimensional piston problem, a mixed implicit/explicit (IMEX) time integration scheme is employed: the implicit scheme is used to integrate the fluid and structural dynamics, whereas an explicit Runge-Kutta scheme integrates the coupling terms. The resulting IMEX scheme retains the order of the implicit and explicit schemes. In the IMEX scheme considered, the implicit scheme consists of an explicit first stage, singly diagonally implicit Runge-Kutta (ESDIRK) scheme, which is a multi-stage, L-stable scheme.

Fourth-Order Runge–Kutta Schemes for Fluid Mechanics Applications

Multiple high-order time-integration schemes are used to solve stiff test problems related to the Navier---Stokes (NS) equations. The primary objective is to determine whether high-order schemes can displace currently used second-order schemes on stiff NS and Reynolds averaged NS (RANS) problems, for a meaningful portion of the work-precision spectrum. Implicit---Explicit (IMEX) schemes are used on separable problems that naturally partition into stiff and nonstiff components. Non-separable problems are solved with fully implicit schemes, oftentimes the implicit portion of an IMEX scheme. The convection---diffusion-reaction (CDR) equations allow a term by term stiff/nonstiff partition that is often well suited for IMEX methods. Major variables in CDR converge at near design-order rates with all formulations, including the fourth-order IMEX additive Runge---Kutta (ARK2) schemes that are susceptible to order reduction. The semi-implicit backward differentiation formulae and IMEX ARK2 schemes are of comparable efficiency. Laminar and turbulent aerodynamic applications require fully implicit schemes, as they are not profitably partitioned. All schemes achieve design-order convergence rates on the laminar problem. The fourth-order explicit singly diagonally implicit Runge---Kutta (ESDIRK4) scheme is more efficient than the popular second-order backward differentiation formulae (BDF2) method. The BDF2 and fourth-order modified extended backward differentiation formulae (MEBDF4) schemes are of comparable efficiency on the turbulent problem. High precision requirements slightly favor the MEBDF4 scheme (greater than three significant digits). Significant order reduction plagues the ESDIRK4 scheme in the turbulent case. The magnitude of the order reduction varies with Reynolds number. Poor performance of the high-order methods can partially be attributed to poor solver performance. Huge time steps allowed by high-order formulations challenge the capabilities of algebraic solver technology.

A higher-order time integration algorithm for the simulation of nonlinear fluid–structure interaction

In this paper higher-order time integration schemes are applied to nonlinear fluid–structure interaction (FSI) simulations. For a given accuracy, we investigate the efficiency of higher-order time integration schemes compared to lower-order methods. In the partitioned FSI simulations on a one-dimensional piston problem, a mixed implicit/explicit (IMEX) time integration scheme is employed: the implicit scheme is used to integrate the fluid and structural dynamics, whereas an explicit Runge–Kutta scheme integrates the coupling terms. The resulting IMEX scheme retains the order of the implicit and explicit schemes. In the IMEX scheme considered, the implicit scheme consists of an explicit first stage, singly diagonally implicit Runge–Kutta (ESDIRK) scheme, which is a multi-stage, L-stable scheme.

Iterative solution techniques for unsteady flow computations using higher order time integration schemes

AbstractIn this paper iterative techniques for unsteady flow computations with implicit higher order time integration methods at large time steps are investigated. It is shown that with a minimal coding effort the standard non‐linear multigrid method can be combined with a Newton–Krylov method leading to speed‐ups in the order of 30%. Copyright © 2005 John Wiley & Sons, Ltd.

Implicit and explicit higher order time integration schemes for structural dynamics and fluid-structure interaction computations

In this paper higher order time integration schemes are applied to structural dynamics and fluid-structure interaction (FSI) simulations. So far only second order accurate time integration schemes have been successfully applied to fluid-structure interaction simulations. For equal accuracy the higher order time integration schemes can require less computational work then a lower order method, leading to a higher computational efficiency. In the partitioned FSI simulations on a one-dimensional piston test problem, a mixed implicit/explicit (IMEX) time integration scheme is employed: the implicit scheme is used to integrate the fluid and structural dynamics, whereas an explicit Runge–Kutta scheme integrates the coupling terms. The resulting IMEX scheme retains the order of the implicit and explicit schemes. In the IMEX scheme considered, the implicit scheme consists of an explicit first stage, singly diagonally implicit Runge–Kutta (ESDIRK) scheme, which is a multi-stage, L-stable scheme. Since, the ESDIRK scheme has not been previously applied to structural dynamics, it is used to integrate a number of simple structural dynamics problems to investigate the performance of the scheme. The ESDIRK scheme allows a direct and efficient integration of the cases considered.

Evaluation of an energy relaxation method for the simulation of unsteady, viscous, real gas flows

AbstractThis study investigates a new energy relaxation method designed to capture the dynamics of unsteady, viscous, real gas flows governed by the compressible Navier–Stokes equations. We focus on real gas models accounting for inelastic molecular collisions and yielding temperature‐dependent heat capacities. The relaxed Navier–Stokes equations are discretized using a mixed finite volume/finite element method and a high‐order time integration scheme. The accuracy of the energy relaxation method is investigated on three test problems of increasing complexity: the advection of a periodic set of vortices, the interaction of a temperature spot with a weak shock, and finally, the interaction of a reflected shock with its trailing boundary layer in a shock tube. In all cases, the method is validated against benchmark solutions and the numerical errors resulting from both discretization and energy relaxation are assessed independently. Copyright © 2004 John Wiley & Sons, Ltd.

Higher-order time integration schemes for the unsteady Navier–Stokes equations on unstructured meshes

The efficiency gains obtained using higher-order implicit Runge–Kutta (RK) schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier–Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each time step are presented. The first algorithm (nonlinear multigrid, NMG) is a pseudo-time-stepping scheme which employs a nonlinear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on inexact Newton’s methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson’s iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the generalized minimal residual method. Results demonstrating the relative superiority of these Newton’s method based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes such as fourth-order Runge–Kutta (RK64) with the more efficient inexact Newton’s method based schemes (LMG).

Concepts and Application of Time-Limiters to High Resolution Schemes

A new class of implicit high-order non-oscillatory time integration schemes is introduced in a method-of-lines framework. These schemes can be used in conjunction with an appropriate spatial discretization scheme for the numerical solution of time dependent conservation equations. The main concept behind these schemes is that the order of accuracy in time is dropped locally in regions where the time evolution of the solution is not smooth. By doing this, an attempt is made at locally satisfying monotonicity conditions, while maintaining a high order of accuracy in most of the solution domain. When a linear high order time integration scheme is used along with a high order spatial discretization, enforcement of monotonicity imposes severe time-step restrictions. We propose to apply limiters to these time-integration schemes, thus making them non-linear. When these new schemes are used with high order spatial discretizations, solutions remain non-oscillatory for much larger time-steps as compared to linear time integration schemes. Numerical results obtained on scalar conservation equations and systems of conservation equations are highly promising.