V.S. Lukin

Spectral element spatial discretization error in solving highly anisotropic heat conduction equation

This paper describes a study of the effects of the overall spatial resolution, polynomial degree and computational grid directionality on the accuracy of numerical solutions of a highly anisotropic thermal diffusion equation using the spectral element spatial discretization method. The high-order spectral element macroscopic modeling code SEL/HiFi has been used to explore the parameter space. It is shown that for a given number of spatial degrees of freedom, increasing polynomial degree while reducing the number of elements results in exponential reduction of the numerical error. The alignment of the grid with the direction of anisotropy is shown to further improve the accuracy of the solution. These effects are qualitatively explained and numerically quantified in 2- and 3-dimensional calculations with straight and curved anisotropy.

Modeling open boundaries in dissipative MHD simulation

The truncation of large physical domains to concentrate computational resources is necessary or desirable in simulating many natural and man-made plasma phenomena. Three open boundary condition (BC) methods for such domain truncation of dissipative magnetohydrodynamics (MHD) problems are described and compared here. A novel technique, lacuna-based open boundary conditions (LOBC), is presented for applying open BC to dissipative MHD and other hyperbolic and mixed hyperbolic-parabolic systems of partial differential equations. LOBC, based on manipulating Calderon-type near-boundary sources, essentially damp hyperbolic effects in an exterior region attached to the simulation domain and apply BC appropriate for the remaining parabolic effects (if present) at the exterior region boundary. Another technique, approximate Riemann BC (ARBC), is adapted from finite volume and discontinuous Galerkin methods. In ARBC, the value of incoming flux is specified using a local, characteristic-based method. A third commonly-used open BC, zero-normal derivative BC (ZND BC), is presented for comparison. These open BC are tested in several gas dynamics and dissipative MHD problems. LOBC are found to give stable, low-reflection solutions even in the presence of strong parabolic behavior, while ARBC are stable only when hyperbolic behavior is dominant. Pros and cons of the techniques are discussed and put into context within the body of open BC research to date.

A priori mesh quality metric error analysis applied to a high-order finite element method

Characterization of computational meshs quality prior to performing a numerical simulation is an important step in insuring that the result is valid. A highly distorted mesh can result in significant errors. It is therefore desirable to predict solution accuracy on a given mesh. The HiFi/SEL high-order finite element code is used to study the effects of various mesh distortions on solution quality of known analytic problems for spatial discretizations with different order of finite elements. The measured global error norms are compared to several mesh quality metrics by independently varying both the degree of the distortions and the order of the finite elements. It is found that the spatial spectral convergence rates are preserved for all considered distortion types, while the total error increases with the degree of distortion. For each distortion type, correlations between the measured solution error and the different mesh metrics are quantified, identifying the most appropriate overall mesh metric. The results show promise for future a priori computational mesh quality determination and improvement.