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| -rw-r--r-- | tutorials/module_3/6_pde.md | 26 |
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diff --git a/tutorials/module_3/6_pde.md b/tutorials/module_3/6_pde.md index 1f11b13..95d63c4 100644 --- a/tutorials/module_3/6_pde.md +++ b/tutorials/module_3/6_pde.md @@ -64,6 +64,9 @@ $$ Explicit methods +Forward-time central-space + based on forward euler method and central difference in space. + Simple Implicit methods Crank-Nicolson ADI @@ -71,7 +74,30 @@ ADI ### Hyperbolic Equations MacCormack Method +In computational fluid dynamics (CFD), the governing equations are the Navier-Stokes equations. For inviscid (no viscosity) compressible flow, these reduce to the Euler equations: +$$ +\frac{\partial u}{\partial t}+\frac{\partial f(u)}{\partial x} = 0 +$$ +where, $U$ is conserved variables. This equation is a hyperbolic PDE. + +Discretize domain: space and time + + + +Write fluxes: + + +MacCormack Algorithm +- Predictor +$$ +u^p_i = u^n_i - \frac{\Delta t}{\Delta x}(f^n_{i+1}-f^n_i) +$$ +- Corrector +$$ +u^{p+1}_i = \frac{1}{2}(u^n_i+u^p_i) - \frac{\Delta t}{2\Delta x}(f^p_{i}-f^n_{i-1}) +$$ +Method of characteristics ## Finite-Element Method General Approach |