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

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.