lecture38 - Lecture 38 Insulated Boundary Conditions...

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Lecture 38 Insulated Boundary Conditions Insulation In many of the previous sections we have considered fixed boundary conditions, i.e. u (0) = a , u ( L ) = b . We implemented these simply by assigning u j 0 = a and u j n = b for all j . We also considered variable boundary conditions, such as u (0 , t ) = g 1 ( t ). For example, we might have u (0 , t ) = sin( t ) which could represents periodic heating and cooling of the end at x = 0. A third important type of boundary condition is called the insulated boundary condition. It is so named because it mimics an insulator at the boundary. Physically, the effect of insulation is that no heat flows across the boundary. This means that the temperature gradient is zero, which implies that should require the mathematical boundary condition u ( L ) = 0. To use it in a program, we must replace u ( L ) = 0 by a discrete version. Recall that in our discrete equations we usually have L = x n . Recall from the section on numerical derivatives, that there are three different ways to replace a derivative by a difference equation, left, right and central differences. The three of them at x n would be: u ( x n ) u n - u n 1 h u n +1 - u n h u n +1 - u n 1 2 h . If x n is the last node of our grid, then it is clear that we cannot use the right or central difference, but are stuck with the first of these. Setting that expression to zero implies: u n = u n 1 . This restriction can be easily implemented in a program simply by putting a statement u(n+1)=u(n) inside the loop that updates values of the profile. However, since this method replaces u ( L ) = 0 by an expression that is only accurate to first order, it is not very accurate and is usually avoided. Instead we want to use the most accurate version, the central difference. For that we should have u ( L ) = u ( x n ) = u n +1 - u n 1 2 h = 0 .
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