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 (2 p- 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.