Efficient predictor/multi-corrector algorithms based on high-order accurate time-discontinuous Galerkin methods for second-order hyperbolic systems

For stiff dynamic systems, e.g. structural dynamics and wave problems governed by second-order hyperbolic equations, time-integration methods with properties of unconditional stability and asymptotic annihilation of high frequency response are needed. Time-discontinuous Galerkin (TDG) methods provide a variational framework for developing high-order accurate approximations with these desirable properties. In this dissertation, an efficient predictor/multi-corrector algorithm for solving the large system of 2(p + 1) neq x 2(p + 1)n eq discrete matrix equations resulting from the TDG method is presented. Here, p is the order of polynomial approximations and neq is the number degree-of-freedom arising from the spatial discretization. The algorithm overcomes the demanding storage and computational effort of the direct solution of the TDG matrix equations. The success of the block matrix iterative solution method is a result of a specific form of time approximation functions. An analysis shows that the algorithm needs only a few iteration passes with a single matrix factorization of size equal to the number of spatial degree-of-freedom neq to reach the same 2p + 1 order of accuracy of the parent TDG solution obtained from a direct solver. For linear (p = 1) and quadratic (p = 2) approximations in time, each iteration pass retains the desirable unconditional and asymptotic stability properties of the parent TDG solution.; To improve the stability of the iterative algorithm based on the TDG formulation, gradient least-squares operators are developed. The method is referred to as time-discontinuous Galerkin/gradient-least-squares (TDG/GLS). Unconditional stability and convergence of the parent TDG/GLS method are proved by functional analysis techniques. The stability and accuracy properties based on finite difference analyses are also presented for both TDG/GLS parent and iterative algorithms. Specific choice of the gradient least-squares parameter is essential to allow the predictor/multi-corrector algorithm extending to the higher-order approximations (p ≥ 3) while retaining the desirable unconditional stability of the TDG/GLS method. The numerical solution obtained from the first pass of the TDG/GLS iterative algorithm is investigated. The results obtained from finite difference analyses show that, with an exception of external loading function, the single-pass TDG/GLS algorithm is equivalent to the stiffly accurate Singly-Diagonal-Implicit Runge-Kutta (SDIRK) methods. The embedded technique for error estimates based on the single-pass algorithm is developed for adaptive time-stepping strategy.; The gradient least-squares operators increase the stability but reduce the super-convergence of the TDG solution at the end of the time step by one order. A sub-multi-pass iterative algorithm for the TDG method with higher-order polynomial approximations (p ≥ 3) is developed to improve the stability condition without degrading the accuracy of the TDG solution. Implementation of the predictor/multi-corrector algorithm based on the TDG formulation to the problem in non-linear dynamics is also presented. The use of purely modified Newton-Raphson iteration reduces the number of factorizations for the Jacobian matrix to one for each time interval. Several numerical examples are presented to demonstrate the efficiency and accuracy of the predictor/multi-corrector algorithms based on the TDG methods over second-order accurate single-step/single-solve (SS/SS) methods.