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High-order computations

 

I analysed the effiiciency of high-order computation for the solution of wave propagation problems and fluid problems. I carried out the comparison using the Continuous Galerkin method and HDG. 

The comparison criteria that I used were:

  • the dimension of the linear system produced by the spatial discretization;

  • the total time to compute the solution, including: the elemental computations, the matrix assembly, the linar system resolution with a direct solver. For HDG, the time to compute the super-convergent postprocess is also taken into account.

 

In the first test case I computed the propagation of a plane wave in a homogeneus medium with the Helmholtz equation. The wave is scatterd by a circular object and for symmetry reason only half of the domain is computed.  The case set up and the soution, for a typical wavelength, are shown below.

The results of the comparison are shown in the following figures. The first comparison criterion is the dimension of the linear system (here called Number of DOF). The CDG method is also considered for the comparison. For a given accuracy, the number of DOF are plotted as a function of the wavenumber of the incoming wave. The upper figures refere to an error computed as L^2 error in the whole domain, while in the bottom figures the error in the scattering boundary is considered. Using this comparison creterion, high-order elements are clearly superior to low-order elements, since they always need less DOF for a fixed accuracy.

The next figures are based on the second comparison criterion, that is, the total time to obtain a solution with a given accuracy. Also in this case, high-order elements are superior to low-orders, but the optimal polynomial degree depends on the accuracy sought and on the wavelength. HDG and CG performs similarly. More examples can be found in my thesis and publications.

A similar comparison has been carried out for the solution of the Navier-Stokes equations. The test case is about the evaluation of the aerodynamics characteristics of a NACA0012 profile with HDG. This comparison is obtained taking four nested meshes with polynomial degree p=1, 2, 4, 8. This four meshes have then the same number of nodes. In the figures below are show, in the right, the p=4 mesh, and in the left a zoom on the leading edge of the meshes p=2, 4, 8.

Using the same criteria as in the case of the wave problem, the four interpolations are compared. The results are shown below with the errors in the aerodynamic characteristics. Very good accuracy is obtained with the p = 4 computation, outperforming linear and quadratic elements. An acceptable accuracy is also obtained with the
p = 8 computation, despite the very coarse discretization.

In the context of HDG for the incompressible Navier-Stokes equations, I developed a fractional-step method for the time integration of the equations. The time advancement is decomposed into a sequence of two or more steps. In particular, the momentum equation is solved in the first step without accounting for the diverge free constrain, obtaining in this way an intermediate velocity that is projected into a divergence free space in the second step. This formulation allows to circumvent the difficulties caused bythe saddle-point nature of the variational formulation of the Navier-Stokes equations,where the pressure variable acts as a Lagrange multiplier of the incompressibility constrain.

In the following example, the Taylor-Green vortex problem at Re= 1600 is solved in a cubic domain (10x10x10) with p = 4 hexaedral elements. From a given initial condition, the flow evolves to a turbulence breakdown, with the creation of small scales, followed by adecay phase similar to decaying homogeneous turbulence. Here, iso-surfaces at value 0.25 of the z-component of thevorticity are shown.

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