Laplacian test cases with Dirichlet boundary conditions in the 2D axisymmetric coordinate system.
Laplacian test cases with Dirichlet boundary conditions in the 2D axisymmetric coordinate system.
The aim of this test case is to validate the discretization of the diffusive term of a cell advection/diffusion equation in the 2D axisymmetric coordinate system.
Domain is rectangular, its opposite corner coordinates being \((0,0)\) and \((1,1)\).
We solve \(-\Delta T=f\) on the whole domain.
Using cylindrical coordinates, 2D solution of the problem is set to \(\cos(\pi r)\sin(\pi y)\). Right-end-side f is equal to \(2\pi^{2}\cos(\pi r)\sin(\pi y) + \dfrac{\pi}{r}\sin(\pi r)\sin(\pi y)\).
The symmetry axis is in located on the left at \( r=0.5 \). The global boundary conditions are chosen as follows.
Boundary | Condition | Value |
---|---|---|
Top | Dirichlet | \( T = 0 \) |
Bottom | Dirichlet | \( T = 0 \) |
Other Boundaries | Neumann | \( \partial T = 0 \) |
Energy equation is used:
Second order spatial convergence is expected.
The grid size starts from 40 in every direction to 640. Second order convergence is observed with \(L_\infty\), \(L_1\) or \(L_2\) norms.
2D results
Mesh | Temperature Linf error | Order | Temperature L1 error | Order | Temperature L2 error | Order |
---|---|---|---|---|---|---|
40 | 8.55732636e-04 | n/a | 2.69430910e-04 | n/a | 3.52892413e-04 | n/a |
80 | 2.14068698e-04 | +1.9991 | 6.73389943e-05 | +2.0004 | 8.81932983e-05 | +2.0005 |
160 | 5.35256476e-05 | +1.9998 | 1.68319461e-05 | +2.0002 | 2.20464624e-05 | +2.0001 |
320 | 1.33819414e-05 | +1.9999 | 4.20792332e-06 | +2.0000 | 5.51149922e-06 | +2.0000 |
640 | 3.34551808e-06 | +2.0000 | 1.05197011e-06 | +2.0000 | 1.37786755e-06 | +2.0000 |