1.1.8.2. Discretising the Euler Equations

Using a Central Difference formula for a vector function:

\[{{\partial \mathbf{F}(x_i)} \over \partial x} = {{\mathbf{F}(x_{i+1}) - \mathbf{F}(x_{i-1})} \over 2 \Delta x} = {{\mathbf{F}_{i+1} - \mathbf{F}_{i-1}} \over 2 \Delta x}\]

Simply replaced scalar u with vector \(\mathbf{F}\)

In all our schemes, similarly replace the scalar derivative:

\[A = {{d F} \over {d u}}\]

By the Jacobian matrix:

\[\mathbf{A} = {{d \mathbf{F}} \over {d \mathbf{U}}}\]

1.1.8.2.1. Lax-Friedrichs

FTCS n to n+1, with spatial average for u at n

\[\mathbf{U}_i^{n+1} = {1 \over 2}(\mathbf{U}_{i+1}^n + \mathbf{U}_{i-1}^n) + {\Delta t \over {2 \Delta x}} ({\mathbf{F}_{i+1}^n - \mathbf{F}_{i-1}^n})\]

1.1.8.2.2. Lax-Wendroff

  • Taylor at n+1,

  • First derivative - replace du/dt with -dF/dx and then CS at i

  • Second derivative - replace d/dt (du/dt) with d/dt(-dF/dx),

    • then replace d/dx (-dF/dt) with d/dx(A dF/dx)

    • then outer derivative is CS at i+0.5 and CS at i-0.5

    • inner derivative is CS at i+0.5 and i-0.5

    • Jacobian is average at i+0.5 and i-0.5

\[\mathbf{U}_i^{n+1} = \mathbf{U}_i^n - {\Delta t \over {2 \Delta x}}({\mathbf{F}_{i+1}^n - \mathbf{F}_{i-1}^n}) + {{\Delta t^2} \over {2 \Delta x^2}} \left[ {\mathbf{A}_{i+{1/2}}^n(\mathbf{F}_{i+1}^n - \mathbf{F}_i^n) - \mathbf{A}_{i-{1/2}}^n(\mathbf{F}_i^n - \mathbf{F}_{i-1}^n)} \right]\]

e.g.

\[\mathbf{A}_{i+{1/2}}^n = {1 \over 2}( \mathbf{A}_{i+1}^n + \mathbf{A}_i^n )\]
\[\mathbf{A}_{i-{1/2}}^n = {1 \over 2}( \mathbf{A}_i^n + \mathbf{A}_{i-1}^n )\]

Note that \(\mathbf{A}\) has 9 entries and so evaluating it is expensive

Plus we have Matrix-Vector multiplication - thus this method is expensive.

1.1.8.2.3. Richtmyer

Step 1: Predictor - Lax-Friedrichs: FTCS n to n+1/2, at i=i+1/2, with spatial average for u at n):

\[\mathbf{U}_{i+1/2}^{n+1/2} = {1 \over 2}(\mathbf{U}_{i+1}^n + \mathbf{U}_{i}^n) - {\Delta t \over {2 \Delta x}} ({\mathbf{F}_{i+1}^n - \mathbf{F}_{i}^n})\]

Step 2 (Corrector - Leapfrog: FTCS n to n+1)

\[\mathbf{U}_{i}^{n+1} = \mathbf{U}_i^n - {\Delta t \over {\Delta x}}({\mathbf{F}_{i+1/2}^{n+1/2} - \mathbf{F}_{i-1/2}^{n+1/2}})\]

1.1.8.2.4. MacCormack

Step 1 (Predictor - FTFS, n to n+1, with \(\Delta t\)):

\[\mathbf{\tilde{U}}_i^{n+1} = \mathbf{U}_i^n - {{\Delta t} \over {\Delta x}}({\mathbf{F}_{i+1}^n - \mathbf{F}_i^n})\]

Step 2 (Corrector - FTBS, n+1/2 to n+1, with \(\Delta t/2\)):

\[\mathbf{U}_i^{n+1} = {1 \over 2} (\mathbf{\tilde{U}}_i^{n+1} + \mathbf{U}_i^n) - {{\Delta t} \over {2 \Delta x}}({\mathbf{\tilde{F}}_i^{n+1} - \mathbf{\tilde{F}}_{i-1}^{n+1}})\]