A Finite Element Method with Special Edge-Polynomials for Advection Problems

Abstract

The field of Computational Fluid Dynamics (CFD) is constantly finding new ways to improve simulation results. One of the large challenges in CFD, is advection. Advection dominated flows are governed by hyperbolic equations, which pose severe stability issues. Stabilized methods usually add significant numerical diffusion, which alters the solution. Another issue are discontinuities in solutions, such as found in shock waves and acoustics. Besides solving these issues, conservative properties have to be satisfied at all costs. In order to further improve results, new methods are still developed today. The mathematical tools to do such developments are also further expanded. For instance, in the last decades the branch of differential geometry has steadily grown. This branch of mathematics can be extremely useful, since a clear distinction is made between geometric depend and geometric independent operators. These operators become more in use for numerical methods. Of all the available operators, the Lie derivative is important for advection. In junction with newly developed edge polynomials, a scheme for advection is created. The new edge polynomials are special, as these partly scale with a variable integrated over the cell and partly scale with the values on the cell boundaries. In order to do this, one polynomials integrates to 1 over the cell, while the other polynomials do not contribute, hence integrate to 0 over a cell. This gives several benefits. Conservation is clearly defined in the cells and not in points, hence conservation is locally defined. Also an extinction is made between conservative values and fluxes. One can also use the property of not contributing to conservation by correcting point values for stabilization or improvement of the solution. These polynomials are used within a Galerkin Finite Elements (FE) framework. Besides correction methods, upwind methods can be used as well. The approach is tested on several equation sets. On a simple linear advection problem, excellent results are found. Four test cases are tried, from a sine wave, two discontinuous functions and a hat function. With the proposed polynomials, the shapes are conserved extremely well, also in comparison to widely used schemes. On the inviscid Burgers’ equations, the stabilization methods become of importance. In the inviscid Burgers’ equation, initially smooth problems can become discontinuous. When a discontinuity forms, several issues can occur. For one, spurious oscillations can become apparent, or the velocity of the discontinuity can have errors. With several approaches, competitive results can be obtained. The discontinuity velocity is computed well, but spurious oscillation usually do occur. With several methods, the oscillations are suppressed. Not all approaches succeed in doing so. The third and final equation set to be tested, are the Euler equations. Although solutions can be found, all solutions contain errors. Several error sources are indicated, which should be addressed in further research. In comparison to a Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) scheme, results need to be further improved to be on par. In conclusion, the suggested polynomials show potential. Several methods to improve solutions showed their usefulness. In linear advection, the new polynomials performed better compared to existing widely used methods. In non-linear advection, the results are on par with reference methods. For the Euler equations, further improvements are necessary. With further