Improvements to Solve Diffusion-Dominant Problems with PFEM-2
Abstract
Particle Finite Element Method - Second generation (PFEM-2) is a method characterized by using both particles and mesh to solve physics equations.
In some previous papers the mathematical and numerical basis of the method with also some results were presented showing a good accuracy and high performance for solving scalar-transport and momentum-transport problems with dominant advection term.
To preserve accurate solutions on the mesh, the method must create or remove particles to have enough information to solve diffusive terms. In these problems and under certain types of velocity fields and/or mesh, some cautions at the particle-updating step must be taken into account to avoid an excessive numerical diffusion in the solutions. In this way, an improved approach to create and remove particles is presented here.
Moreover, when the problem is diffusion-dominant, the advantages of the method are not as clear as in the advection-dominant cases. The explicit calculation of the diffusion traditionally used by PFEM-2 is limited by the dimensionless Fourier number and, in some particular cases, the method may fail. To relax these restrictions, a new model to calculate the diffusion is developed. It is based on the theta method which consists on discretizing the non-stationary variable using a weighted mixture between an explicit prediction and an implicit correction. This approach extends the stability limit, while reducing the error, and it could also allow the usage of longer time steps. Even though the implicit may be computationally expensive, exploiting the possibility of factorizing the matrix at the beginning and using it as a preconditioner for solving the diffusion step had shown to have good performance.
To validate the solutions proposed here some tests are solved comparing their different results with the analytic ones. This report is focused on showing how the new models improve the accuracy of the solutions of the older ones and how the computing times are modified.
In some previous papers the mathematical and numerical basis of the method with also some results were presented showing a good accuracy and high performance for solving scalar-transport and momentum-transport problems with dominant advection term.
To preserve accurate solutions on the mesh, the method must create or remove particles to have enough information to solve diffusive terms. In these problems and under certain types of velocity fields and/or mesh, some cautions at the particle-updating step must be taken into account to avoid an excessive numerical diffusion in the solutions. In this way, an improved approach to create and remove particles is presented here.
Moreover, when the problem is diffusion-dominant, the advantages of the method are not as clear as in the advection-dominant cases. The explicit calculation of the diffusion traditionally used by PFEM-2 is limited by the dimensionless Fourier number and, in some particular cases, the method may fail. To relax these restrictions, a new model to calculate the diffusion is developed. It is based on the theta method which consists on discretizing the non-stationary variable using a weighted mixture between an explicit prediction and an implicit correction. This approach extends the stability limit, while reducing the error, and it could also allow the usage of longer time steps. Even though the implicit may be computationally expensive, exploiting the possibility of factorizing the matrix at the beginning and using it as a preconditioner for solving the diffusion step had shown to have good performance.
To validate the solutions proposed here some tests are solved comparing their different results with the analytic ones. This report is focused on showing how the new models improve the accuracy of the solutions of the older ones and how the computing times are modified.
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ISSN 2591-3522