A balanced adaptive time-stepping strategy is implemented in an implicit discontinuous Galerkin solver to guarantee the temporal accuracy of unsteady simulations.A proper relation between the spatial,temporal and iter...A balanced adaptive time-stepping strategy is implemented in an implicit discontinuous Galerkin solver to guarantee the temporal accuracy of unsteady simulations.A proper relation between the spatial,temporal and iterative errors generated within one time step is constructed.With an estimate of temporal and spatial error using an embedded RungeKutta scheme and a higher order spatial discretization,an adaptive time-stepping strategy is proposed based on the idea that the time step should be the maximum without obviously infuencing the total error of the discretization.The designed adaptive time-stepping strategy is then tested in various types of problems including isentropic vortex convection,steady-state fow past a fat plate,Taylor-Green vortex and turbulent fow over a circular cylinder at Re=3900.The results indicate that the adaptive time-stepping strategy can maintain that the discretization error is dominated by the spatial error and relatively high efciency is obtained for unsteady and steady,well-resolved and under-resolved simulations.展开更多
The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomia...The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials,modified to accommodate a C~0-continuous expansion. Computationally and theoretically, by increasing the polynomial order p,high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed.展开更多
基金Zhen-Guo Yan acknowledges supports from the National Natural Science Foundation of China(Grant no.11902344)National Numerical Windtunnel Project.The development of the implicit solver in Nektar++has been supported by EPSRC grant(EP/R029423/1)UK Turbulence Consortium grant(EP/R029326/1).
文摘A balanced adaptive time-stepping strategy is implemented in an implicit discontinuous Galerkin solver to guarantee the temporal accuracy of unsteady simulations.A proper relation between the spatial,temporal and iterative errors generated within one time step is constructed.With an estimate of temporal and spatial error using an embedded RungeKutta scheme and a higher order spatial discretization,an adaptive time-stepping strategy is proposed based on the idea that the time step should be the maximum without obviously infuencing the total error of the discretization.The designed adaptive time-stepping strategy is then tested in various types of problems including isentropic vortex convection,steady-state fow past a fat plate,Taylor-Green vortex and turbulent fow over a circular cylinder at Re=3900.The results indicate that the adaptive time-stepping strategy can maintain that the discretization error is dominated by the spatial error and relatively high efciency is obtained for unsteady and steady,well-resolved and under-resolved simulations.
文摘The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials,modified to accommodate a C~0-continuous expansion. Computationally and theoretically, by increasing the polynomial order p,high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed.
