 CFSD at TU München | Flashcards & Summaries

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TESTE DEIN WISSEN

Parabolic PDE

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

Elliptic partial differential equations (PDE)

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TESTE DEIN WISSEN
• typically model steady-state or equilibrium phenomena
•  Typical equation is the Poisson equation
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TESTE DEIN WISSEN

Typical ill-posed problem

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TESTE DEIN WISSEN

inverse heat conduction

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TESTE DEIN WISSEN

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

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TESTE DEIN WISSEN

due to numerical instability

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TESTE DEIN WISSEN

conservative

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TESTE DEIN WISSEN

mimics a global conservation law

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TESTE DEIN WISSEN

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

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TESTE DEIN WISSEN

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

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TESTE DEIN WISSEN

dispersion relation

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The modified phase velocity is a function of the wavenumber itself

This phenomenon is called dispersion relation of the discretization scheme.

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TESTE DEIN WISSEN

Lax-Equivalence Theorem

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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.

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TESTE DEIN WISSEN

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?

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TESTE DEIN WISSEN

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

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TESTE DEIN WISSEN

Concept of linear stability fails for

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

Typical well-posed problem

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TESTE DEIN WISSEN

heat conduction

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TESTE DEIN WISSEN

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

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TESTE DEIN WISSEN

• a solution exists,

• the solution is unique,

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

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## Beispielhafte Karteikarten für deinen CFSD Kurs an der TU München - von Kommilitonen auf StudySmarter erstellt!

Q:

Parabolic PDE

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

Elliptic partial differential equations (PDE)

A:
• typically model steady-state or equilibrium phenomena
•  Typical equation is the Poisson equation
Q:

Typical ill-posed problem

A:

inverse heat conduction

Q:

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

A:

due to numerical instability

Q:

conservative

A:

mimics a global conservation law

Q:

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

A:

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

Q:

dispersion relation

A:

The modified phase velocity is a function of the wavenumber itself

This phenomenon is called dispersion relation of the discretization scheme.

Q:

Lax-Equivalence Theorem

A:

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.

Q:

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?

A:

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

Q:

Concept of linear stability fails for

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

Typical well-posed problem

A:

heat conduction

Q:

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

A:

• a solution exists,

• the solution is unique,

• the solution depends continuously on the data (initial and boundary conditions) ### Erstelle und finde Lernmaterialien auf StudySmarter.

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