This test case solves the sole diffusion equation with explicit and implicit schemes.

Configurations

A cubic cavity of unit length is used to test advection schemes in the 3 directions. Explicit schemes are integrated with a second order RK scheme and implicit schemes with a second order BDF.

Runtime parameters

Main parameters to run this test case are:

• diffusion schemes: "o2_centered","o4_centered"
• directions: [1,2,3]
• Fourier number: [0.01,1.0]

Results

Pure sinus

Figure 1: Diffusion of a pure sinus with various schemes. Exact solution (in gray w/ crosses), explicit second order centered (o2_centered, in red), explicit fourth order centered (o4_centered, in orange), implicit second order centered w/ first order Euler (o2_centered w/ time_order_discretization 1, in light green w/ triangles, outside of the plot's range), implicit second order centered w/ second order BDF (o2_centered w/ time_order_discretization 2, in olive w/ triangles), implicit second order centered w/ third order BDF (o2_centered w/ time_order_discretization 3, in turquoise w/ triangles).

Analysis

We confirm that the first and second order implicit BDF schemes, coupled with the second order centered scheme, are less diffusive than expected. On the contrary, the third order BDF scheme is more diffusive. The explicit schemes, coupled with RK2, are also less diffusive than the exact solution, with a smaller error for the fourth order scheme.

References