Changes between Version 7 and Version 8 of u/erica/truncerror

Timestamp:

08/28/13 14:47:42 (12 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/truncerror

@@ -67 +67 @@
 = A check that numerically solving Laplace's Eqn with f(x) = 6x gives the exact solution = 
 
+I wanted to check if numerically solving 
 
+[[latex($\triangledown ^2 U = 6x$)]] 
+
+gives the exact solution. This is the same as saying 
+
+[[latex($A\vec{U} = 6\vec{x}$)]],
+
+that is the LTE is zero (i.e. discretization of the equation leads to the exact solution).
+
+To check this I wrote out the form for the LTE, which is the same as given above but with f(xi) replaced with 6x(i). Using the exact solution in this formula gives,
+
+[[latex($\tau_i = U''_i + \frac{1}{12}h^2 U''''_i + O(h^4) - 6x_i$)]]
+
+The ODE is easy to solve by hand, and gives a general solution of, 
+
+[[latex($U(x) = x^3 + cx + d$)]]
+
+As you can see, the truncation error requires 2nd derivatives and ''higher'', and so the LTE is exactly zero as I wanted to verify.
+
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