Changes between Version 6 and Version 7 of u/erica/truncerror

Timestamp:

08/28/13 14:40:35 (11 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/truncerror

@@ -35 +35 @@
 Replacing terms with their Taylor series indicates, 
 
-[[latex($\frac{1}{h^2}[h^2 U''_i + \frac{1}{12}h^4 u''''_i + \cdot \cdot \cdot]-f(x_i) = U''_i + O(h^2) - f(x_j) = \tau _i$)]]
+[[latex($\frac{1}{h^2}[h^2 U''_i + \frac{1}{12}h^4 U''''_i + \cdot \cdot \cdot]-f(x_i) = U''_i + \frac{1}{12}U''''_i(h^2) +O(h^4) - f(x_j) = \tau _i$)]]
+
+Using the original ODE, this becomes, 
+
+[[latex($\tau_i = \frac{1}{12}U''''_i  + O(h^4)$)]]
+
+= Global error = 
+
+As we had the matrix formula for the finite difference system of equations, 
+
+[[latex($A\vec{u} = \vec{x}$)]], 
+
+analogously when we replace the approximate solution u with the true solution, U, we have to introduce the local truncation error, 
+
+[[latex($A\vec{U} = \vec{x} + \tau$)]]
+
+Taking the difference of these equations, we have 
+
+[[latex($A\vec{E} = - \tau$)]]
+
+This is a set of finite difference equations for the global error. 
+
+Using reverse logic, we can assume this system is an approximation to the ODE:
+
+[[latex($\frac{d^2E}{dx^2}=-\tau(x)$)]]
+
+with boundary conditions E(0)=E(mx+1)=0 for the case of Dirchlet BCs (where value of u is supplied at boundary).
+
+Integration of this equation is said to provide an estimate for the global error in your solution, which you'd like to be bound and decreasing for increasing h.
+
+= A check that numerically solving Laplace's Eqn with f(x) = 6x gives the exact solution = 
+
+