Changes between Version 5 and Version 6 of u/erica/truncerror

Timestamp:

08/28/13 14:25:34 (11 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/truncerror

@@ -29 +29 @@
 where [[latex(${\vec{u}}$)]] is the solution vector from this finite-difference scheme. 
 
-Now if we replace the approximate solution with the exact solution in the discrete equation above, the left and right hand sides in general will not be exactly the same. This difference is the LTE, and can be written as, 
+Now if we replace the approximate solution with the exact solution (U) in the discrete equation above, the left and right hand sides in general will not be exactly the same. This difference is the LTE, and can be written as, 
 
-[[latex($\tau = \frac{1}{h^2}[u(x_i+h)+u(x_i-h)-2u(x_i)] - f(x_i)$)]]
+[[latex($\tau = \frac{1}{h^2}[U(x_i+h)+U(x_i-h)-2U(x_i)] - f(x_i)$)]]
+
+Replacing terms with their Taylor series indicates, 
+
+[[latex($\frac{1}{h^2}[h^2 U''_i + \frac{1}{12}h^4 u''''_i + \cdot \cdot \cdot]-f(x_i) = U''_i + O(h^2) - f(x_j) = \tau _i$)]]