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CFSD

Elliptic partial differential equations (PDE)

- typically model steady-state or equilibrium phenomena
- Typical equation is the Poisson equation

CFSD

Parabolic PDE

- diffusion phenomena, i.e. the time-dependent solution of an elliptic problem
- Heat conduction

CFSD

Any system of differential equations describes a well-posed problem for the given initial and boundary conditions given that

• a solution exists,

• the solution is unique,

• the solution depends continuously on the data (initial and boundary conditions)

CFSD

Typical well-posed problem

heat conduction

CFSD

Typical ill-posed problem

inverse heat conduction

CFSD

Numerical solution of a well-posed problem can be look like ill-posed if..?

due to numerical instability

CFSD

Loss of theoretical advantages of FEM in CFD due to...

Comparably complex nonlinearities occur in the compressible Euler equations and in the Navier-Stokes equations

CFSD

Numerical schemes need to pass tests and then validated for more general cases under smoothness assumption. WHat are the numerical test equations and testing concepts?

Our numerical test equations are:

o Linear advection equation

o Burgers equation

o 1D Euler equation

Testing concepts of:

o Stability

o Consistency

o Convergence

CFSD

Lax-Equivalence Theorem

Given a well-posed initial-value problem for a linear scalar PDE and a linear discretization scheme of the form (VI.3) that is consistent, LR-stability is necessary and sufficient for convergence.

CFSD

dispersion relation

The modified phase velocity is a function of the wavenumber itself

This phenomenon is called dispersion relation of the discretization scheme.

CFSD

Concept of linear stability fails for

- Gibbs oscillations caused by Discontinuities
- Regions of sharp solution gradients for which similar reasoning applies
- Compressible flows can develop shocks

CFSD

conservative

mimics a global conservation law

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