High-order Time-stepping Schemes Research Articles

A Novel Gauged Potential Formulation for 3-D Electromagnetic Field Analysis Including Both Inductive and Capacitive Effects

In this paper, a novel potential formulation for low-frequency (LF) applications taking into account both inductive and capacitive effects but without considering wave propagation is proposed. Both time-domain and frequency-domain formulations are presented. The resultant fully discrete finite-element matrix is made symmetric by incorporating a gauge condition and also rewriting the current continuity equation. To improve numerical accuracy and computational efficiency, high-order mixed edge elements and nodal elements are adopted to approximate the vector and scalar unknown variables together with high-order time-stepping schemes. Several numerical examples are solved to validate and showcase the accuracy of the proposed methods. The proposed formulations are stable in use for LF electromagnetic field computations by considering inductive and capacitive effects simultaneously, such as finding the resonant frequencies of wireless power transfer devices.

Arbitrarily high-order time-stepping schemes based on the operator spectrum theory for high-dimensional nonlinear Klein–Gordon equations

In this paper we explore arbitrarily high-order Lagrange collocation-type time-stepping schemes for effectively solving high-dimensional nonlinear Klein–Gordon equations with different boundary conditions. We begin with one-dimensional periodic boundary problems and first formulate an abstract ordinary differential equation (ODE) on a suitable infinity-dimensional function space based on the operator spectrum theory. We then introduce an operator-variation-of-constants formula which is essential for the derivation of our arbitrarily high-order Lagrange collocation-type time-stepping schemes for the nonlinear abstract ODE. The nonlinear stability and convergence are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi-definite matrix under some suitable smoothness assumptions. With regard to the two dimensional Dirichlet or Neumann boundary problems, our new time-stepping schemes coupled with discrete Fast Sine / Cosine Transformation can be applied to simulate the two-dimensional nonlinear Klein–Gordon equations effectively. All essential features of the methodology are present in one-dimensional and two-dimensional cases, although the schemes to be analysed lend themselves with equal to higher-dimensional case. The numerical simulation is implemented and the numerical results clearly demonstrate the advantage and effectiveness of our new schemes in comparison with the existing numerical methods for solving nonlinear Klein–Gordon equations in the literature.

Code verification for unsteady 3-D fluid–solid interaction problems

This paper describes a procedure to synthesize Manufactured Solutions for Code Verification of an important class of Fluid–Structure Interaction (FSI) problems whose behaviors can be modeled as rigid body vibrations in incompressible fluids. We refer this class of FSI problems as Fluid–Solid Interaction problems, which can be found in many practical engineering applications. The methodology can be utilized to develop Manufactured Solutions for both 2-D and 3-D cases. We demonstrate the procedure with our numerical code. We present details of the formulation and methodology. We also provide the reasonings behind our proposed approach. Results from grid and time step refinement studies confirm the verification of our solver and demonstrate the versatility of the simple synthesis procedure. In addition, the results also demonstrate that the modified decoupled approach to verify flow problems with high-order time-stepping schemes can be employed equally well to verify code for multi-physics problems (here, those of the Fluid–Solid Interaction) when the numerical discretization is based on the Method of Lines.

Time‐accurate solution of stabilized convection–diffusion–reaction equations: I—time and space discretization

AbstractThe paper addresses the development of time‐accurate methods for solving transient convection–diffusion–reaction problems using finite elements. Multi‐stage time‐stepping schemes of high accuracy are used. They are first combined with a Galerkin formulation to briefly recall the time–space discretization. Then spatial stabilization techniques are combined with high‐order time‐stepping schemes. Moreover, a least‐squares formulation is also developed for these high‐order time schemes combined with C0 finite elements (in spite of the diffusion operator and without reducing the strong form into a system of first‐order differential equations). The weak forms induced by the SUPG, GLS, SGS and least‐squares formulations are presented and compared. In a companion paper (Part II of this work), the phase and damping properties of the developed schemes are analysed and numerical examples are included to confirm the effectiveness of the proposed methodology for solving time‐dependent convection–diffusion–reaction problems. Copyright © 2002 John Wiley & Sons, Ltd.

ON OPEN BOUNDARIES IN THE FINITE ELEMENT APPROXIMATION OF TWO-DIMENSIONAL ADVECTION-DIFFUSION FLOWS

SUMMARY A steady-state and transient finite element model has been developed to approximate, with simple triangular elements, the two-dimensional advection—di⁄usion equation for practical river surface flow simulations. Essentially, the space—time Crank—Nicolson—Galerkin formulation scheme was used to solve for a given conservative flow-field. Several kinds of point sources and boundary conditions, namely Cauchy and Open, were theoretically and numerically analysed. Steady-state and transient numerical tests investigated the accuracy of boundary conditions on inflow, noflow and outflow boundaries where di⁄usion is important (di⁄usive boundaries). With the proper choice of boundary conditions, the steady-state Galerkin and the transient Crank—Nicolson—Galerkin finite element schemes gave stable and precise results for advectiondominated transport problems. Comparisons indicated that the present approach can give equivalent or more precise results than other streamline upwind and high-order time-stepping schemes. Di⁄usive boundaries can be treated with Cauchy conditions when the flow enters the domain (inflow), and with Open conditions when the flow leaves the domain (outflow), or when it is parallel to the boundary (noflow). Although systems with mainly di⁄usive noflow boundaries may still be solved precisely with Open conditions, they are more susceptible to be influenced by other numerical sources of error. Moreover, the treatment of open boundaries greatly increases the possibilities of correctly modelling restricted domains of actual and numerical interest. ( 1997 by John Wiley & Sons, Ltd.