Numerical discretizations of partial differential equations, like Navier-Stokes and magnetohydrodynamics, using fixed grid Eulerian methods lead to numerical viscosity, resistivity and dissipation. This is manifested in acoustic and Alfvenic wave damping. Here we show how to derive such dissipation from the truncation error of the numerical method.

1. Numerical Viscosity and Diffusion in
Finite Difference Eulerian Codes
Simon Candelaresi, Dominika Zieba

2. What is Numerical Diffusion?
2
Everyone is talking about it,
but no one knows what is really is.

3. Numerical Experiments
3

4. Wave Damping
4

5. Analytical Approach
5

6. Local Truncation Error
6
discretized exact
This is solved exactly.

7. Numerical Diffusion
7
PDEs:
In the PencilCode do we have ?
What is ?
Approach:
1. Discretize PDEs.
2. Apply method of lines to get set of coupled ODEs.
3. Construct the Runge-Kutta intermediate steps.
4. Eliminate off-center values using the Taylor expansion.
5. Eliminate intermediate time steps using time Taylor expansion.

8. Inviscid Navier-Stokes 3d
Truncation errors with :
Similar for and .
Proper diffusion terms:
8
second order space
second order Runge-Kutta

9. Conclusions
simon.candelaresi@gmail.com
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Numerical viscosity and diffusion can be calculated analytically.
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Need to find proper interpretation of the terms.
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Next: higher order space and time discretization, MHD.