Differential Equations Lecture Work Solutions 320

Differential Equations Lecture Work Solutions 320 - 9.7...

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9.7 Matrix Method for Stability 9.8 Derivative Boundary Conditions 9.9 Hyperbolic Equations 9.9.1 Stability Problems 1. Use a von Neumann stability analysis to show for the wave equation that a simple explicit Euler predictor using central diFerencing in space is unstable. The diFerence equation is u n +1 j = u n j c t ax u n j +1 u n j 1 2 ! Now show that the same diFerence method is stable when written as the implicit formula u n +1 j = u n j c t x u n +1 j +1 u n +1 j 1 2 ! 2. Prove that the C±L condition is the stability requirement when the Lax WendroF method is applied to solve the simple 1-D wave equation. The diFerence equation is of the form: u n +1 j = u n j c t 2∆ x u n j +1 u n
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