CFD lecture 8 - Computational Fluid Dynamics Lecture 8...

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Computational Fluid Dynamics Lecture 8 Combining Time and Space Differencing 0 uu C tx ∂∂ += 1. Suppose forward time and centered space unstable. 2. Leap Frog and centered space neutral, conditionally stable 3. Forward time and one sided space could be stable, conditionally stable 4. Leap Frog and one-sided space unstable, because L-F has waves going both ways. It is possible that damping from one sided spatial differences with amplification from a time discretization to produce a stable scheme. Consider L-F and Central differences, L-F has positive phase speed error, and Central differences have negative (decelerating) phase speed errors. Sometimes it is useful to examine time and space differencing properties together. We use a traveling wave solution of the form: () with r i r ik j x n t n j ri nt n j n n j e i ee Ae ω φ ωω ∆− = =+ = = Determining the imaginary part of is equivalent to a Von Neumann stability analysis. Phase speed properties are contained in r . 0 nd 11 Examine L-F and 2 Central differences 0 22 nn jj A C φφ +− = −− ∆∆ Substitute in wave solution 1
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() i 22 or sin sin Discrete sin sin Dispersion Relation where If 1 will be real and the scheme is neutral. 0
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CFD lecture 8 - Computational Fluid Dynamics Lecture 8...

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