A high-order full-discretization method using Hermite interpolation for periodic time-delayed differential equations

A high-order full-discretization method （FDM） using Hermite interpolation （HFDM） is proposed and implemented for periodic systems with time delay. Both Lagrange interpolation and Hermite interpolation are used to approximate state values and delayed state values in each discretization step. The transition matrix over a single period is determined and used for stability analysis. The proposed method increases the approximation order of the semidiscretization method and the FDM without increasing the computational time. The convergence, precision, and efficiency of the proposed method are investigated using several Mathieu equations and a complex turning model as examples. Comparison shows that the proposed HFDM converges faster and uses less computational time than existing methods.

An ultra-accurate dynamic isogeometric analysis with higher order mass formulation

An ultra-accurate isogeometric dynamic analysis is presented.The key ingredient of the proposed methodology is the development of isogeometric higher order mass matrix.A new one-step method is proposed for the construction of higher order mass matrix.In this approach,an adjustable mass matrix is formulated through introducing a set of mass parameters into the consistent mass matrix under the element mass conservation condition.Then the semi-discrete frequency derived from the free vibration equation with the adjustable mass matrix is served as a measure to optimize the mass parameters.In 1D analysis,it turns out that the present one-step method can perfectly recover the existing reduced bandwidth mass matrix and the higher order mass matrix by choosing different mass parameters.However,the employment of the proposed one-step method to the2D membrane problem yields a remarkable gain of solution accuracy compared with the higher order mass matrix generated by the original two-step method.Subsequently a full-discrete isogeometric transient analysis algorithm is presented by using the Newmark time integration scheme and the higher order mass matrix.The full-discrete frequency is derived to assess the accuracy of space-time discretization.Finally a set of numerical examples are presented to evaluate the accuracy of the proposed method,which show that very favorable solution accuracy is achieved by the present dynamic isogeometric analysis with higher order mass formulation compared with that obtained from the standard consistent mass approach.

Two-grid stabilized mixed finite element method for fully discrete reaction-diffusion equations

Two-grid mixed finite element method is proposed based on backward guler schemes for the unsteady reaction-diffusion equations. The scheme combines with the stabilized mixed finite element scheme by using the lowest equal-order pairs for the velocity and pressure. The space twogrid method is also used to reduce the time consuming. The benefits of this approach are to avoid the higher derivative, but to have more favorable stability, and to get the numerical solution of the two unknown variables simultaneously. Stability analysis and error estimates are given in this work. Finally, the theoretical results are verified by the numerical examples.

Space-Time Discontinuous GalerkinMethod for Maxwell’s Equations

A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell’s equations.Distinguished from the Runge-Kutta discontinuous Galerkin method(RKDG)and the finite element time domain method(FETD),in our scheme,discontinuous Galerkinmethods are used to discretize not only the spatial domain but also the temporal domain.The proposed numerical scheme is proved to be unconditionally stable,and a convergent rate O((△t)^(r+1)+h^(k+1/2))is established under the L^(2)-normwhen polynomials of degree atmost r and k are used for temporal and spatial approximation,respectively.Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction.An ultra-convergence of order(△t)^(2r+1) in time step is observed numerically for the numerical fluxes w.r.t.temporal variable at the grid points.