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Given

$(\textrm{Pe}-1)u_{i+1} + 2u_i - (\textrm{Pe}-1)u_{i-1} = \frac{h^2}{\mu}$

where $\textrm{Pe}$ stands for Péclet number, $h$ is the step size of the mesh and $\mu$ is the diffusion (constant)

How to solve such equation? I'm trying to solve it to understand why there are no oscillations in a finite element method approximation with the upwind schema.

UPDATE 1

The roots I got are: $r_1 = 1$ and $r_2 = \frac{1+\textrm{Pe}}{1-\textrm{Pe}}$

So I have the following guess for the general solution

$u_i = C_1r_1^i + C_2r_2^i$

To find the constants $C_1$ and $C_2$ I need to initial conditions. What are those? Are they related to the boundary conditions? I have $u(0) = 1$ and $u'(1) = 1$. How to proceed?

UPDATE 2

According to this text, I replace the roots in my guess for the solution to the homogeneous part.

$u_i = C_1(1)^i + C_2\Big(\frac{1+\textrm{Pe}}{1-\textrm{Pe}}\Big)^i$

Applying the initial conditions I have

$u_0 = 1 = C_1(1)^0 + C_2\Big(\frac{1+\textrm{Pe}}{1-\textrm{Pe}}\Big)^0$ $1-C_1 = C2$

for the other initial condition, what should I do, I mean I don't know how to proceed with that Neumann condition there.

UPDATE 3

Before applying the initial conditions to find the values of $C_1$ and $C_2$ I tried to solve for the particular case following the example on the information of this site

So I came up with using a guess like $f_i = ki$. Replacing on the original equation I have

$(\textrm{Pe}-1)k(i+1) + 2ki - (\textrm{Pe}-1)k(i-1) = \frac{h^2}{\mu}$

which after rearranging gives me

$\big[(\textrm{Pe}-1)+2-(\textrm{Pe}-1)\big]ki + \big[(\textrm{Pe}-1)+(\textrm{Pe}+1)\big]k = \frac{h^2}{\mu}$

finally $k = \frac{h^2}{\mu \big[(\textrm{Pe}-1)+(\textrm{Pe}+1)\big]}$

Hence, the complete (I think) equation is:

$u_i = C_1(1)^i + C_2\Big(\frac{1+\textrm{Pe}}{1-\textrm{Pe}}\Big)^i+\frac{h^2 i}{\mu \big[(\textrm{Pe}-1)+(\textrm{Pe}+1)\big]}$

Am I right? I just need to apply initial conditions. The $u(0) = 1$ is easy but the other one $u'(1) = 1$ how to apply that?

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    Oh! and please someone tell me (if possible) if this the wa$y$ I should be analyzing to know why there are no oscillations on the numerical approximation of this problem.2012-12-19

1 Answers 1

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It depends. Do A and B depend on the index $i$? If A and B are constant, then you can split the solution up into a homogeneous solution $u_{i}^{(H)}$ and an inhomogeoneous solution $u_{i}^{(I)}$. The homogeneous solution satisfies

$A u_{i+1}^{(H)} + 2 u_{i}^{(H)} + B u_{i-1}^{(H)} = 0 $

with initial conditions such as $u_{0}^{(H)} = u_0$ and $u_{1}^{(H)} = u_1$. In this case, with constant coefficients, you can assume that $u_{i}^{(H)}$ takes the form $\exp{(i r)}$ and find $r$ from the following equation:

$A r^2 + 2 r + B = 0$

which I leave to you; the solutions are then $r_1$ and $r_2$, and the general solution to the homogeneous equation is $u_{i}^{(H)} = C_1 \exp{(i r_1)} + C_2 \exp{(i r_2})$. $C_1$ and $C_2$ are found by using the initial conditions.

The inhomogeneous solution satisfies

$A u_{i+1}^{(I)} + 2 u_{i}^{(I)} + B u_{i-1}^{(I)} = C $

with initial conditions such as $u_{0}^{(I)} = 0$ and $u_{1}^{(I)} = 0$. For a constant $C$, $u_{i}^{(I)}$ can be a constant, with such constant being offset in the homogeneous initial conditions.

For the case where $A$ and/or $B$ is not constant, then methods specific to the forms of their variation should be employed.

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    I have edited my post to give more context. I'll check your answer and see if that helps, though.2012-12-19