From 65bccf39525578eb0c763c3dd8ec4e4c9f9a7306 Mon Sep 17 00:00:00 2001
From: Christian Kolset <christian.kolset@gmail.com>
Date: Tue, 23 Sep 2025 15:43:39 -0600
Subject: Added FTCS method to PDE.md

---
 tutorials/module_3/6_pde.md | 26 ++++++++++++++++++++++++++
 1 file changed, 26 insertions(+)

(limited to 'tutorials')

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
-- 
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