Generalized Petrov-Galerkin time finite element weighted residual methodology for designing high-order unconditionally stable algorithms with controllable numerical dissipation

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Abstract

In this paper, a novel stabilized time-weighted residual methodology under the umbrella of Petrov-Galerkin time finite element formulation is developed to design a generalized computational framework, which permits unconditionally stable, high-order time accuracy, and features with controllable numerical dissipation for solving transient first-order systems. Various unconditionally stable (A/L-stable) algorithms can be readily obtained in the proposed framework having not only high-order accuracy but also controllable numerical dissipation in the high frequency. Quadratic and cubic basis functions are utilized to illustrate the specific design process of the proposed method, which consequently ends up with numerous third-/fifth-order time accurate algorithms, QUAD3(γ,ρ) and CUBE5(γ,ρ), with controllable numerical dissipation. Comparing with the well-known implicit Runge-Kutta (RK) family of algorithms, these newly developed QUAD3(γ,ρ) and CUBE5(γ,ρ) can (a) recover the Radau IIA3/RK3 and Radau IIA5/RK5 schemes by certain selection of algorithmic parameters (γ,ρ); (b) obtain numerous new algorithms with improved solution accuracy that is superior to the RK family of algorithms, and (c) has similar computational efficiency as that of the implicit RK family of algorithms. Several single/multi-degree of freedom (SDOF and MDOF) problems are investigated to validate the proposed developments. In addition, it is worth noting that the proposed methodology can also use high-order (not limited to the quadratic and cubic) basis functions to design more advanced schemes and can be integrated with high-order spatial discretization methods in the solution of space-time PDEs, such as isogeometric methods, SEM, Discontinuous Galerkin (DGM), p-versions FEM, etc.

Introduction

Temporal discretized operators are fundamental and crucial for transient simulations in terms of predicting the physics in engineering applications, such as fluid flow, heat transfer and structural dynamics. In general, unconditionally stable algorithms are preferred as they can use a large time step to obtain stable results with a significant payoff regarding the computational cost; whereas, in the case with large stiffness, high-frequency components lead to physically incorrect and/or numerically oscillatory results and may blow up the simulations. To eliminate those nonphysical and/or numerical oscillations, the time discretization algorithms are usually expected to have controllable numerical dissipation. Different numerical approximations, encompassing finite difference method [1], [2], [3], [4], generalized time-weighted residual method [5], [6], [7], finite element formulations in space and time [8], [9], [10], and the like, have been utilized to deriving computational frameworks for time discretization, such as linear multi-step methods, θ-method (generalized Trapezoidal family of algorithms, or generalized-α method for first-order systems), and Runge-Kutta-type methods, etc. It has been pointed out [11], [12], [13], [14], [15], [16], [17], [18] that most of these algorithms can be recast in a unified manner, that is to derive them from a generalized time-weighted residual methodology. Inspired by this generalization concept, this exposition starts from the generalized time-weighted procedure, while integrating with Petrov-Galerkin time finite element method, to propose a unified computational framework. That is, we can develop and design high-order unconditionally stable techniques for temporal discretization, and additionally with controllable numerical oscillation behavior in high frequency.

Regarding existing first-order time accurate algorithms, the unconditionally stable Euler backward algorithm is the only L-stable method in the generalized Trapezoidal family, showing excellent dissipation features and yielding smooth results with a large time step Δt (in high frequency). However, the Euler backward algorithm is limited by its low order of accuracy and may be insufficient for practical applications. The only second-order algorithm in the generalized Trapezoidal family is the Crank-Nicolson method, which has no dissipation and shows oscillatory behavior in the high frequency. In terms of second-order time accurate algorithms with controllable numerical dissipation, the Generalized Single-Step Single-Solve family of algorithms for transient first-order systems (GS4-1) and second-order systems (GS4-2) developed by Prof. Tamma's research group [11], [12], [13], [14], [15], [16], [17], [18] show a general representation encompassing the entire class of LMS methods involving a single-step and a single-solve representation within each time step. In GS4-1, two free parameters named principle root ρ and spurious root ρs are introduced to control the numerical damping in high frequency. Specifically, ρ=ρs=1 recovers the Crank-Nicolson method without numerical dissipation; ρ=ρs=0 recovers Gear's method with the greatest numerical damping; 0<ρsρ<1 generates new algorithms with selectively controllable numerical dissipation. However, we cannot get higher-order (greater than 2) A/L stable algorithms with this GS4-1 structure according to Dahlquist first barrier theorem [19] (also see the generalized barrier theorem by Zhou and Tamma (Ref. [5])).

A possible technique to formulate high-order time integration algorithms is introducing the concept of Bubnov Galerkin theory to approximate the time dimension. Similar to that in space finite element methods, high-order basis function usually leads to high-order numerical accuracy. The time weighted residual method was initially proposed by Zienkiewicz and Taylor in [4] for the Single-step p,j method and Generalized Newmark p,j methods. The specific pros/cons regarding these algorithms were carefully investigated and advanced more generally to the entire GS4-1/-2 family of algorithms in [5], [20]. Regarding applying the Petrov-Galerkin discretization in time, several pioneering studies have been reported in Refs. [21], [22], [23], [24], [25] in which pth-order polynomials were employed for the solution and (p1)th for the test functions and the applications covered both first-order systems (heat transfer problems) and second-order systems (wave propagation problems). Moreover, various time finite element methods can be found in [26], [27], [28], [29] and the references therein. Due to the diversity of the Galerkin methodology, the concept of discontinuous Galerkin [30], [31], [32], [33] is also introduced for improved physics and computational performance under certain situations. Moreover, it is worth noting that some of the time finite element methods are embedded in the framework of space-time system [34], [35], [36], [37], [38], that is to say, both space and time dimensions are consistently discretized and approximated in the manner of Galerkin theory. These classical space-time methods are designed at a system level; in particular, the n-dimensional problem in space is ultimately escalated to (n+1)-dimensional problem, and as such computational efficiency is a drawback inherent to the space-time finite element method. Besides, spectral element methods have also been extended to this space-time element concept [38], [39], [40], [41].

In this paper, we aim to develop a unified framework to design high-order A/L-stable algorithms in the realm of the stabilized time-weighted method. To this end, we apply the fundamental Petrov-Galerkin theory to the time dimension that inherently provides us a wide design range of algorithms. All the desired algorithmic properties (‘wish list’): unconditionally stable, pth-order accuracy, and controllable numerical dissipation, are imposed as constraint equations to the free algorithmic parameters via the concept of Algorithms by Design [5], [6], [7], [42], [43], leading to a unified framework to design advanced time finite element framework. Quadratic and cubic basis functions are utilized as illustrative examples to present design details for third-/fifth-order unconditionally stable algorithms with controllable numerical dissipation features; moreover, second-/fourth-order algorithms can be designed with different accuracy constraints as well. In addition, this generalized Petrov-Galerkin time finite element framework can also permit other basis functions for higher-order accurate algorithms with controllable numerical dissipation, and can be readily integrated with higher-order spatial discretization techniques to formulate space-time methodologies with arbitrarily high orders of accuracy in space and time.

Section snippets

Petrov-Galerkin time-weighted residual methodology

Consider the following general semi-discretized linear first-order transient system with initial conditions:{Cu˙(t)+Ku(t)=f(t)u(t=0)=u0 where u˙=ut denotes the first derivative of u upon time t. C and K are system matrices generated by space discretization techniques, such as finite/spectral element method (FEM/SEM), finite volume method (FVM), finite difference method (FDM), isogeometric analysis, Discontinuous Galerkin method (DGM), and the like. Firstly, we cast Eq. (1) into a generalized

Design of algorithm structure

Considering a homogeneous problem with single degree of freedom, u˙+λu=0. Applying the quadratic basis functions to design algorithms under the umbrella of the Petrov-Galerkin time-weighted residual framework, we have the following design details:

Step 1

As illustrated in Fig. 1, choose three points, [tn,tn+γ,tn+1], as the interpolation points. The time increment t[tn,tn+1] is firstly mapped to an isoparametric element of s[1,1] by:s=2tn+1tn(ttn)1

Step 2

The generalized quadratic basis functions are

Design example-II: Cubic basis functions leading to fifth-order accurate algorithms with controllable numerical dissipation

In this section, we give a brief design procedure via using cubic basis function and show the parameters for fifth-order accurate algorithms with controllable numerical dissipation. As illustrated in Fig. 3, choose four points, [tn,tn+γ1,tn+γ2,tn+1], as the interpolation points, and therefore the generalized cubic basis functions are defined as:N0=(s1)(s2γ1+1)(s2γ2+1)8γ1γ2N1=(s1)(s+1)(s2γ2+1)8γ1(γ11)(γ1γ2)N2=(s1)(s+1)(s2γ1+1)8γ2(γ21)(γ2γ1)N3=(s+1)(s2γ1+1)(s2γ2+1)8(γ11)(γ21)

Single degree of freedom problem

In this section, we first consider the following single degree of freedom (SDOF) problem:u˙+λu=10cos(t/10)u0=100 where the exact solution is given by:u(t)=(u01000λ1+100λ2expλt)+100[10λcos(t/10)+sin(t/10)]1+100λ2 and the numerical error is defined as:Error=|NumericalExactExact|

For the QUAD3(γ,ρ) family of algorithms, Fig. 5 presents the convergence rate plots for different selections of (γ,ρ). It can be found that all algorithms with different selections of (γ,ρ) show third-order

Conclusions

In this paper, a stabilized Petrov-Galerkin time finite element weighted residual methodology was proposed to preserve unconditional stability, high-order accuracy, and features with controllable numerical dissipation for the time discretization. Different from existing time finite element methods, the proposed unified computational framework introduced a stabilized term that achieves controllable numerical dissipation of high-frequency components. Quadratic and cubic basis functions were

CRediT authorship contribution statement

Yazhou Wang: Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. Kumar K. Tamma: Conceptualization, Methodology, Project administration, Supervision, Writing – original draft, Writing – review & editing. Tao Xue: Formal analysis, Methodology, Validation. Dean Maxam: Formal analysis, Methodology, Validation. Guoliang Qin: Funding acquisition, Project administration, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 51776155). The author Yazhou Wang would like to thank for the financial support from China Scholarship Council (No. 201906280340). Thanks are also due to Prof. Tamma's computational mechanics laboratory at the University of Minnesota.

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