Controllable Numerical Dissipation Research Articles

Algebraically stable SDIRK methods with controllable numerical dissipation for first/second-order time-dependent problems

In this paper, a family of four-stage singly diagonally implicit Runge-Kutta methods are proposed to solve first-/second-order time-dependent problems, exhibiting the following numerical properties: fourth-order accuracy in time, unconditional stability, controllable numerical dissipation, and adaptive time step selection. The BN-stability condition is employed as a constraint to optimize parameters in the Butcher table, having significant benefits, and hence is recommended for nonlinear dynamics problems in contrast to existing methods. Numerical examples involving both first- and second-order linear/nonlinear dynamics problems validate the proposed method, and numerical results reveal that the proposed methods are free from the order reduction phenomenon when applied to nonlinear dynamics problems. The performance of adaptive time-stepping using the embedded scheme is further illustrated by the phase-field modeling problem. Additionally, the advantages and disadvantages of three-stage third-order accurate algebraically stable methods are discussed. The proposed high-order time integration can be readily integrated into high-order spatial discretization methods, such as the high-order spectral element method employed in this paper, to obtain high-order discretization in space and time dimensions.

The generalization of diagonally implicit Runge–Kutta–Nyström method with controllable numerical dissipation for structural dynamics

The generalization of diagonally implicit Runge–Kutta–Nyström method with controllable numerical dissipation for structural dynamics

A robust and generalized effective DAE framework encompassing different methods, algorithms, and model order reduction for linear and nonlinear second order dynamical systems

For linear and nonlinear dynamical problems, we propose the novel development and advancement of the well-known Generalized Single Step Single Solve (GS4) family of second order time-accurate algorithms encompassing the entire class of LMS methods developed over the past 50 years or so with/without controllable numerical dissipation in conjunction with the Differential Algebraic Equation (DAE) framework and Proper Orthogonal Decomposition (POD). Unlike traditional practices which have severe limitations in the ability to provide generality and flexibility and a wide variety of choices to the analyst without losing order of time accuracy, the proposed implementation uniquely not only allows altogether different numerical time integration algorithms within the GS4 family in each of the different subdomains in a single body, but also additionally allows the selection of different space discretized methods such as the Finite Element Method (FEM) and particle methods, and other spatial methods as well in a single analysis. Moreover, the addition of the POD further provides reduction in computational times. Furthermore, unlike existing state of the art, the present framework readily permits a wide array of implicit–implicit, implicit–explicit, and explicit–explicit couplings and the integration of such a technology is of interest here. Consequently, the present DAE-GS4-POD framework has the flexibility of using different spatial methods and different time integration schemes in different subdomains in a selective manner together with Reduced Order Modeling (ROM) to optimize capturing the local and global features of the representative physics. The ROM also employs an iterative convergence check in acquiring sufficient snapshot data to adequately capture the physics to the prescribed accuracy requirements. Such a novelty, and computational features, are not possible with existing state of art and numerical illustrations validate the claim.

High-order implicit time integration scheme with controllable numerical dissipation based on mixed-order Padé expansions

A single-step high-order implicit time integration scheme with controllable numerical dissipation at high frequencies is presented for the transient analysis of structural dynamic problems. The amount of numerical dissipation is controlled by a user-specified value of the spectral radius ρ∞ in the high frequency limit. Using this user-specified parameter as a weight factor, a Padé expansion of the matrix exponential solution of the equation of motion is constructed by mixing the diagonal and sub-diagonal expansions. An efficient time-stepping scheme is designed where systems of equations, similar in complexity to the standard Newmark method, are solved recursively. It is shown that the proposed high-order scheme achieves high-frequency dissipation, while minimizing low-frequency dissipation and period errors. The effectiveness of the provided dissipation control and the efficiency of the scheme are demonstrated by numerical examples. A simple guideline for the choice of the controlling parameter and time step size is provided. The source codes written in MATLAB and FORTRAN are available for download at: https://github.com/ChongminSong/HighOrderTimeIntegration.

Higher-Order Accurate Explicit Time Schemes with Improved Dissipation Properties

In this paper, a novel family of fourth-order accurate explicit time integration schemes is developed by combining the new time approximations and the explicit fourth-order Runge–Kutta (RK4) method. Inaccurate predictions due to the presence of excessive numerical dissipation are often observed in practical analyses of structural dynamics when the RK4 is employed for both displacement and velocity approximations. To remedy this, novel time approximations with adjustable algorithmic parameters are employed for the displacement vectors while the velocity vectors are approximated by using the RK4. For the complete elimination of numerical dissipation, algorithmic parameters are unconventionally determined by taking the determinant of the amplification matrix as unity. A set of algorithmic parameters obtained from this process makes the new schemes completely non-dissipative while keeping the computational cost the same as the RK4. Due to this improvement, the new schemes have enhanced total energy-conserving capabilities for nonlinear systems and give noticeably more accurate predictions in practical analyses. Until now, controllable numerical dissipation and fourth-order accuracy are not attained simultaneously in a unified set of time schemes. In the new schemes, however, a systematic way to adjust the level of numerical dissipation is also presented while attaining fourth-order accuracy. The numerical results of various test problems show that the new time schemes can provide more accurate predictions for nonlinear conservative dynamic problems than the existing time schemes.

An integrated enabling technology interfacing multiple space/time methods/algorithms/domains with model reduction for first-order systems

PurposeThe purpose of this study is to further advance the multiple space/time subdomain framework with model reduction. Existing linear multistep (LMS) methods that are second-order time accurate, and useful for practical applications, have a significant limitation. They do not account for separable controllable numerical dissipation of the primary variables. Furthermore, they have little or no significant choices of altogether different algorithms that can be integrated in a single analysis to mitigate numerical oscillations that may occur. In lieu of such limitations, under the generalized single-step single-solve (GS4) umbrella, several of the deficiencies are circumvented.Design/methodology/approachThe GS4 framework encompasses a wide variety of LMS schemes that are all second-order time accurate and offers controllable numerical dissipation. Unlike existing state-of-art, the present framework permits implicit–implicit and implicit–explicit coupling of algorithms via differential algebraic equations (DAE). As further advancement, this study embeds proper orthogonal decomposition (POD) to further reduce model sizes. This study also uses an iterative convergence check in acquiring sufficient snapshot data to adequately capture the physics to prescribed accuracy requirements. Simple linear/nonlinear transient numerical examples are presented to provide proof of concept.FindingsThe present DAE-GS4-POD framework has the flexibility of using different spatial methods and different time integration algorithms in altogether different subdomains in conjunction with the POD to advance and improve the computational efficiency.Originality/valueThe novelty of this paper is the addition of reduced order modeling features, how it applies to the previous DAE-GS4 framework and the improvement of the computational efficiency. The proposed framework/tool kit provides all the needed flexibility, robustness and adaptability for engineering computations.

Extensions of single-step method for equations of motion from multibody dynamics

• Improved single-step methods are extended to DAEs from multibody dynamics . • Optimal preconditioning strategies are employed in Newton-Raphson iteration. • All the new methods do not suffer from small time-step-size restriction. • Each new method has second-order accuracy and controllable numerical dissipation. Each second-order, unconditionally stable, linear multi-step (LMS) method has its equivalent single-step (SS) method. For the linear two-step method, LMS2, the corresponding SS method was developed as SS2. In this paper, SS2 for first-order ordinary differential equations (ODEs) is designated as SS2-1 and for second-order ODEs is designated as SS2-2. Then SS2-1 and SS2-2 methods are extended to the numerical integration of index 3 differential-algebraic equations (DAEs), index 2 DAEs, and GGL-stabilized index 2 DAEs from multibody dynamics. Optimal preconditioning strategies are employed in the new methods for DAEs. The right and left preconditioners cure the sensitivity to the solution perturbation and the conditioning of the Jacobian matrix for Newton-Raphson iteration. The extended methods are validated by numerical experiments. The effectiveness of the optimal preconditioning strategy and the violation characteristics of constraint equations for the new methods are also verified. The new methods have the advantages of second-order accuracy, and controllable numerical dissipation and they do not suffer from the small time-step-size restriction. Further, there is no acceleration accuracy reduction for equations of motion (EoM) with a non-constant mass matrix. The efficiency of the new methods and the existing generalized-α method for index 3 DAEs are also compared.

A suite of second-order composite sub-step explicit algorithms with controllable numerical dissipation and maximal stability bounds

This paper constructs a composite s-sub-step explicit method and analyzes second-order accuracy, conditional stability, and dissipation control at the bifurcation point. In the present s-sub-step method, each explicit scheme achieves identical second-order accuracy for analyzing general structures and provides maximal stability bound, that is 2×s where s denotes the number of sub-steps. Except for the single-sub-step case, each explicit scheme achieves dissipation control at the bifurcation point. After registering second-order accuracy and controllable numerical dissipation, the composite multi-sub-step explicit method should be well-designed to reach maximal stability bound. The analysis reveals that as the number of sub-steps increases, the developed explicit schemes can reduce numerical low-frequency dissipation and enlarge stability. Under the same computational cost, the advantage of reducing low-frequency dissipation and enlarging stability is gradually weakened with the increase of sub-steps, so the first seven explicit schemes are only developed and compared in this paper. Some typical experiments are provided to confirm the methods’ numerical performance. The proposed explicit schemes are more accurate and efficient for some models than existing second-order algorithms of that class.

A truly self-starting composite isochronous integration analysis framework for first/second-order transient systems

Due to the lack of and limited work done for first/second-order systems useful for multidisciplinary problems, this paper proposes a new and novel composite isochronous integration ([i-Integration]) analysis framework to solve first/second-order transient systems via a single computational framework. The main contributions are summarized as follows: (1) The ρ∞-Bathe method is newly reformed to an identical truly self-starting representation with the absence of acceleration and is more efficient for practical applications; (2) The novel [i-Integration] process is applied to automatically generate a family of time integration algorithms for first/second-order transient systems, whilst preserving features of second-order time accuracy, unconditional stability, controllable numerical dissipation in high-frequency, and truly self-starting; (3) The newly proposed algorithms design framework provides a unified toolkit to solve first/second-order transient systems in a single analysis and hence is more effective for coupled problems and numerical analysis of fluid/heat transfer/structure-type transient simulations. Numerical examples encompassing a manufactured example, the C-F heat conduction model, and the thermal stress wave problem are demonstrated to validate the proposed algorithms.

An efficient hybrid implicit time integration method with high accuracy for linear and nonlinear dynamics

In this work, an effective hybrid implicit method is proposed to analyze linear and nonlinear dynamic problems. The proposed hybrid method is a type of three sub-step method and is unconditionally stable. New method has multiple free parameters which can be employed to control accuracy and numerical dissipation. The new method can obtain controllable numerical dissipation which ranges from the non-dissipative case to the asymptotic annihilating case. Meanwhile, high computation accuracy than classical sub-step methods can be obtained. The new method can obtain the identical effective stiffness matrix for three sub-steps to save computational cost and internal storage for linear cases. Verified by numerical examples, the new method has simple calculation procedure, and is efficient for linear dynamics. Especially for nonlinear dynamics, the new method shows high solution accuracy and excellent energy conservation performance when compared with other methods.

Adaptive numerical dissipation control for high-order k-exact reconstruction schemes on vertex-centered unstructured grids using artificial neural networks

Due to their enhanced numerical dissipation properties, high-order discretization methods are an important prerequisite to obtain accurate results with Large-Eddy Simulations. However, the exact amount of dissipation often requires a careful tuning by the user via problem-dependent parameters. In this work we present a fully adaptive dissipation control, which ensures stability and additionally reduces the numerical dissipation to a minimum. This novel approach employs a simple feedforward neural network model, which indirectly tabulates an underlying stability equation and thus reduces the computational overhead to estimate the dissipation during runtime. The methodology is adapted for a high-order k-exact reconstruction method on fully unstructured vertex-centered grids, and it is implemented in a full production flow solver. Based on several test cases, the enhanced accuracy compared to a conventional low-order scheme is demonstrated. Especially when dealing with Large-Eddy Simulation benchmarks, significant savings in computation time and grid resolution requirements can be obtained for reaching a desired level of accuracy. Moreover, compared to a high-order reconstruction method with constant numerical dissipation, the presented adaptive approach consistently yields accurate results, regardless of the flow problem.

Numerical dissipation control of a hybrid large-particle method in vortex instability problems

Current trends in the development of numerical schemes are associated with a decrease in dissipative and dispersion errors, as well as an improvement in the grid convergence of the solution. Achieving the computational properties is not an easy problem since a decrease in scheme viscosity is often associated with an increase in the oscillations of gas dynamic parameters. The paper presents a study of the issues of numerical dissipation control in gas dynamics problems in order to increase the resolution in numerical reproduction of vortex instability at contact boundaries. To solve this problem, a hybrid large-particle method of the second order of approximation in space and time on smooth solutions is used. The method is constructed with splitting by physical processes in two stages: gradient acceleration and deformation of the finite volume of the medium; convective transfer of the medium through its facets. An increase in the order of approximation in time is achieved by a time correction step. The regularization of the numerical solution of problems at the first stage of the method consists in the nonlinear correction of artificial viscosity which, regardless of the grid resolution, tends to zero in the areas of smoothness of the solution. At the stage of convective transport, the reconstruction of fluxes was carried out by an additive combination of central and upwind approximations. A mechanism for regulating the numerical dissipation of the method based on a new parametric limiter of artificial viscosity is proposed. The optimal adjustment of the method by the ratio of dissipative and dispersive properties of the numerical solution is achieved by setting the parameter of the limiting function. The efficiency of the method was tested on two-dimensional demonstrative problems. In one of them, the contact surfaces are twisted into a spiral on which the Kelvin-Helmholtz vortex instability develops. Another task is the classic problem with a double Mach reflection of a strong shock wave. Comparison with modern numerical schemes has shown that the proposed variant of the hybrid large-particle method has a high competitiveness. For example, in the problem with double Mach reflection, the considered version of the method surpasses in terms of vortex resolution the popular WENO (Weighted Essentially Non-Oscillatory) scheme of the fifth order and is comparable to the numerical solution of WENO of the ninth order of approximation. The proposed method can be the basis of a convective block of a numerical scheme when constructing a computational technology for modeling turbulence.

A hybrid self-adjustable explicit–explicit–implicit time marching formulation for wave propagation analyses

In this paper, a novel hybrid explicit/implicit time marching procedure is proposed for wave propagation analyses. The new formulation considers local self-adjustable time integration parameters that are defined for each element of the adopted spatial discretization procedure, following the properties of the discretized model. Thus, the explicit/implicit subdomains of the analysis are established at an element level, taking into account automated computations. Two types of explicit approaches are considered here, allowing extended stability limits to be provided and more efficient analyses performed. Controllable numerical dissipation is also enabled following the locally defined adaptive time integrators of the method. In this sense, optimally designed time integration parameters are developed, providing null spectral radii for the highest sampling frequency of the elements, and close to one values for the important low-frequency response domain. Thus, the influence of spurious high-frequency modes is very effectively extinguished and reduced dissipative and dispersive errors are introduced into the analysis. At the end of the paper numerical results are provided, illustrating the excellent performance of the proposed new methodology.

Two third-order explicit integration algorithms with controllable numerical dissipation for second-order nonlinear dynamics

No literature has reported an explicit integration algorithm to achieve controllable numerical dissipation and identical third-order accuracy simultaneously. This paper develops two novel explicit integration algorithms to achieve this goal well without increasing computational complexity. In detail, two novel methods are identical third-order accuracy, so avoiding the order reduction for solving damped and forced vibrations, and both of them provide a full range of dissipation control via adjusting their unique algorithmic parameter. Apart from these two highlights, two novel methods embed another two enjoyably advantages. One is to present a significantly larger stability limit than the published third-order explicit schemes. Two novel methods provide the maximum stability limit, reaching 23 in the non-dissipative case, getting close to four. The other is to maintain a relatively little computational cost. Two novel methods require explicit solutions only twice within each time step. Therefore, two novel methods are significantly superior to other composite sub-step explicit schemes with respect to the accuracy, stability, dissipation control, and computational cost. Numerical examples are also performed to confirm the numerical performance and superiority of two novel explicit methods.

A linear stability analysis of compressible hybrid lattice Boltzmann methods

An original spectral study of the compressible hybrid lattice Boltzmann method (HLBM) on standard lattice is proposed. In this framework, the mass and momentum equations are addressed using the lattice Boltzmann method (LBM), while finite difference (FD) schemes solve an energy equation. Both systems are coupled with each other thanks to an ideal gas equation of state. This work aims at answering some questions regarding the numerical stability of such models, which strongly depends on the choice of numerical parameters. To this extent, several one- and two-dimensional HLBM classes based on different energy variables, formulations (primitive or conservative), collision terms and numerical schemes are scrutinized. Once appropriate corrective terms introduced, it is shown that all continuous HLBM classes recover the Navier-Stokes-Fourier behavior in the linear approximation. However, striking differences arise between HLBM classes when their discrete counterparts are analyzed. Multiple instability mechanisms arising at relatively high Mach number are pointed out and two exhaustive stabilization strategies are introduced: (1) decreasing the time step by changing the reference temperature Tr and (2) introducing a controllable numerical dissipation σ via the collision operator. A complete parametric study reveals that only HLBM classes based on the primitive and conservative entropy equations are found usable for compressible applications. Finally, an innovative study of the macroscopic modal composition of the entropy classes is conducted. Through this study, two original phenomena, referred to as shear-to-entropy and entropy-to-shear transfers, are highlighted and confirmed on standard two-dimensional test cases.

A three-stage explicit time integration method with controllable numerical dissipation

A three-stage explicit time integration method with controllable numerical dissipation

A Comparative Study from Spectral Analyses of High-Order Methods with Non-Constant Advection Velocities

The response of numerical methods to flow properties dynamic changes is a key ingredient of numerical modeling calibration. In this context, a range of spectral analyses and canonical fluid mechanics problems are explored to compare the responses of the high-order spectral difference scheme and of the flux reconstruction methods. Spatial eigen-analysis (Hu et al. in J Comput Phys 151(2):921–946, 1999; Hu and Atkins in J Comput Phys 182(2):516–545, 2002), based on spatially evolving oscillations, is used to get useful insights for problems with inflow/outflow boundary conditions (i.e., non-periodic), while non-modal analysis (Fernandez et al. in Comput Methods Appl Mech Eng 346:43–62, 2019. https://doi.org/10.1016/j.cma.2018.11.027 ) focusses on the short-term dynamics of the discretised system. These two approaches are also extended to the general scalar conservation law with non-constant advection velocity. All these tools are used to obtain more insight about the expected behavior of the mentioned numerical methods for typical engineering applications. On this regard, despite the lack of a mathematical proof of stability in the non-linear case, the selected numerical schemes, for which an extensive literature is available for applications to Euler and Navier–Stokes equations, will be assumed to be stable. Findings are then verified considering one-dimensional linear advection, and two- and three-dimensional flows modeled with the compressible Euler equations. Implications in the accurate control of numerical dissipation for turbulent flows is also addressed. Within the context of high-order methods, this work complement the former temporal spectral analyses of the discontinuous Galerkin approach discretising the linear advection equation.

An Improved Time Integration Method Based on Galerkin Weak Form with Controllable Numerical Dissipation

Based on the newly proposed generalized Galerkin weak form (GGW) method, a two-step time integration method with controllable numerical dissipation is presented. In the first sub-step, the GGW method is used, and in the second sub-step, a new parameter is introduced by using the idea of a trapezoidal integral. According to the numerical analysis, it can be concluded that this method is unconditionally stable and its numerical damping is controllable with the change in introduced parameters. Compared with the GGW method, this two-step scheme avoids the fast numerical dissipation in a low-frequency range. To highlight the performance of the proposed method, some numerical problems are presented and illustrated which show that this method possesses superior accuracy, stability and efficiency compared with conventional trapezoidal rule, the Wilson method, and the Bathe method. High accuracy in a low-frequency range and controllable numerical dissipation in a high-frequency range are both the merits of the method.

A‐stable linear two‐step time integration methods with consistent starting and their equivalent single‐step methods in structural dynamics analysis

AbstractA spectral consistent starting procedure is proposed for the first‐order‐type A‐stable linear two‐step (LTS) time integration methods in structural dynamics analysis. The accuracy analysis for the LTS methods in structural dynamics is presented based on the first‐order model, which enables the algorithms in first‐order transient systems to be extended to structural dynamics under the umbrella of a consistent framework. It is indicated that the approximation of the loads plays an essential role in the equivalence between the LTS methods and some single‐step methods. An algorithmic equivalence feature, which is stricter than the spectral equivalence, is revealed between the two‐leg generalized‐α methods in first‐order systems and the A‐stable LTS methods. The measures of the error constant in the LTS methods and the ultimate spectral radius are extended to optimize a class of single‐step methods of a given convergence order. The spectral consistent starting procedure for the A‐stable LTS methods is developed by utilizing the algorithmic equivalence of the single‐step methods, which results in an optimal A‐stable LTS (OALTS) method possessing a consistent spectral radius with controllable numerical dissipation in the starting step and the steps thereafter. Comparing with the equivalent single‐step methods, the present OALTS method does not need the auxiliary variables, and its displacement, velocity and acceleration can achieve second‐order accuracy simultaneously. The performance of the present OALTS method is verified by numerical examples including physical damping, external loads, and/or nonlinearity.

An Overview of High-Order Implicit Algorithms for First-/Second-Order Systems and Novel Explicit Algorithm Designs for First-Order System Representations

In this paper, we are interested in high-order algorithms for time discretization and focus upon the high-order implicit/explicit algorithm designs. Five high-order unconditionally stable implicit algorithms, derived by the time continuous Galerkin method, weighting parameter method, collocation method, differential quadrature method, and the modified time-weighted residual method, in first-/second-order transient systems are taken into consideration. The present overview and contributions encompass: (1) The pros and cons of various methodologies for the design of high-order algorithms are first demonstrated. Generally, p unknown variables leads to the optimized $$(2p-1)$$ th-order accurate algorithms with controllable numerical dissipation, and/or 2pth-order accurate algorithms without controllable numerical dissipation. (2) Although it is claimed that the TCG method can achieve 2pth-order accuracy with controllable numerical dissipation, it will be shown in this paper that the conclusion was arrived via an inconsistent analysis for the accuracy and the controllable numerical dissipation. (3) Given the rapid increase on the computational cost for high-order algorithms, the iterative predictor/multi-corrector technique is applied to show a novel design for the high-order explicit algorithms derived from the high-order implicit algorithms for the first-order transient systems. Coupled with the high-order Legendre SEM (likewise isogeometric analysis, DG methods, p-version FEM, etc., can be employed) for the spatial discretization, this newly proposed explicit numerical framework can achieve and preserve high-order accuracy in both space and time. In comparison to the famous explicit Runge-Kutta method, these newly designed explicit algorithms have better solution accuracy with comparable stability region.