![]() We are able to calculate more accurate solutions than reported in Marti et al. (2014), implementing the hydrodynamic and magnetohydrodynamic equations. We then run a series of benchmark problems proposed in Marti et al. We propose and study several unit tests which demonstrate the code can accurately solve linear problems, implement boundary conditions, and transform between spectral and physical space. The expansion makes it straightforward to solve equations in tensor form (i.e., without decomposition into scalars). ![]() Nonlinear terms are calculated by transforming from the coefficients of the spectral series to the value of each quantity on the physical grid, where it is easy to calculate products and perform other local operations. The code expands tensorial variables in a spectral series of spin-weighted spherical harmonics in the angular directions and a scaled Jacobi polynomial basis in the radial direction, as described in Vasil et al. ![]() ![]() To aid future comparison to these benchmarks, we include the source code used to generate the data, as well as the data and analysis scripts used to generate the figures.Ībstract = "We present a simulation code which can solve a broad range of partial differential equations in a full sphere. We also demonstrate that in low resolution simulations of the dynamo problem, small changes in a numerical scheme can lead to large changes in the solution. We find the rotating convection and convective dynamo benchmark problems depend sensitively on details of timestepping and data analysis. (2014) by running at higher spatial resolution and using a higher-order timestepping scheme. Also perhaps a simpler example worked out.We present a simulation code which can solve a broad range of partial differential equations in a full sphere. An explanation of how to generally find the divergence of a tensor would be much appreciated. For instance, in my case (for 2D) what are the values of $i$ and $k$ that I should be ranging over? Also what is the meaning of that comma in the index for the tensor - it is not anywhere included in the definition of $T$. ![]() This doesn't really seem to make any sense to me though. A quick google search says that it should be: I wanted to then write out the component-wise equations of $(1)$ but to do that I needed to expand $\nabla\cdot T$ but I honestly have no idea how to do that. I started by writing out the individual components of the tensor $T$ and could pretty easily see that it is symmetric (not sure if this matters). I am working through a fluid dynamics paper and came across this equation: ![]()
0 Comments
Leave a Reply. |