Magnetic Pearson's Vortex

In order to test the code we tried a simple exact solution to the above equations. It is a simple extension of the Pearson's vortex solution for the incompressible Navier-Stokes equations. We used a periodic box, length 1, and set the initial conditions at t=0:

We discretized the domain with a four-by-four array of quadrilaterals as shown in figure 6.1. Also provided are the pressure and magnetic streamfunctions at time t=2. In figure 6.3 we show that the error at time t=2 decays exponentially with increasing expansion order. The parameters used are shown in table 6.2.

  

Figure 6.1: Incompressible magnetic Pearson's vortex (t=2, instantaneous fields). Top: Periodic spectral element mesh, Bottom Left: Pressure field, Bottom Right: Magnetic streamfunction.

  

Figure 6.2: Incompressible magnetic Pearson's vortex. Convergence plot for L2 error in x-component of magnetic field at time t=2 versus expansion order.

 

Parameter Value

Dimension 2d

S_v 100

S_r 100

$$\Delta t$$ 0.001

N-Range 10 to 25

K_Quad 64

Method Galerkin

We also used this test case to determine the time accuracy of the splitting scheme. We fixed the expansion order at N=10 and run the simulation to t=1 for different values of $\Delta t$. The maximum $\Delta t$ was dictated by the CFL condition. In figure 6.3 we show that using the first-order coefficients resulted in a unit slope in the log-log plot of L₂ error versus $\Delta t$. For the second-order coefficients we see second-order convergence.

  

Figure 6.3: Incompressible magnetic Pearson's vortex. Time accuracy plot for the simulation run with N=10. Convergence plot for L2 error in x-component of velocity at time t=1 versus time step Delta t.