FDM Heat Equation FTCS Scheme with NBC

In summary, the conversation discusses using a FTCS scheme with two Essential Boundary Conditions and a Natural Boundary Condition for a PDE. The problem statement and grid spacing are given, and a Finite Difference Approximation is constructed. The conversation also mentions using a central difference approximation for the derivative boundary condition and the need for additional grid lines to satisfy the boundary condition.
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Homework Statement



I have a quiz question that I'm struggling with. We've been working on using a FTCS scheme with two Essential Boundary Conditions, and now I have a problem with one EBC (ie static) and a Natural Boundary Condition (ie a derivative). The condensed problem statement:

[itex]u_{t} = 2u_{xx}, 0 < x < 1, t > 0[/itex]
[itex]u_{x}(0,t) = 0.5[/itex] <- NBC
[itex]u(1,t) = 3[/itex] <- EBC

and grid spacing [itex]Δx = 0.2, Δt = 0.01[/itex]

Construct a Finite Difference Approximation to the PDE using FTCS scheme. Approximate the derivative boundary condition using a central difference approximation at x = 0.

Homework Equations



As above. [itex][/itex]

The Attempt at a Solution



The basic scheme I have been using is this:

[itex]U^{n+1}_{i} = sU^{n}_{i-1} + (1-2s)U^{n}_{i}+sU^{n}_{i+1}[/itex], where [itex]s = \kappa \frac{Δt}{(Δx)^{2}} = 2 \frac{0.01}{0.04} = 0.5[/itex]

With regards to the derivative boundary condition, I believe that because [itex]u_{x}(0,t) = 0.5[/itex] is a positive gradient, that there is heating (not cooling) at the boundary. In my notes I have the following scheme for a Neumann Condition (heat flux at boundary):

Second order central dierence approximation to the boundary condition. Need grid point
outside the boundary to satisfy the boundary condition.


[itex]k \frac{U^{n}_{j+1} - U^{n}_{j-1}}{2Δx} = Fl[/itex] where [itex]k[/itex] is the conductivity and [itex]Fl[/itex] is the heat flux.

If this is even the correct expression, I'm not sure how to integrate it into my FTCS scheme in order to take into account the derivative boundary condition. In class we did some algebraic rearrangement of the equation to get [itex]U^{n}_{j-1}[/itex] by itself (LHS) and then substituted the RHS into the scheme, which seems appropriate:

[itex]U^{n+1}_{i} = \frac{2sΔx.Fl}{k} + (1-2s)U^{n}_{i}+2sU^{n}_{i+1}[/itex]

but we never worked the solution. I'm not sure which values to use for [itex]k[/itex] and [itex]Fl[/itex]. I think one of them will be 0.5? Also, it appears to me that I have now defined a constant heat flux at each point [itex]Δx[/itex] as I compute along the spatial dimension for [itex]U^{n}_{i}, (i=0,1,2,3,4,5) (n = time step)[/itex], but this is just speculation, I'm not a physics/eng major and this is all very new to me.
 
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  • #2
You basically need another grid line at x = -h. Then you have the usual equation relating u at -h, 0, and +h, plus you approximate the derivative with [u(+h, t) - u(-h, t)]/2h = a, where a is whatever constant the x-derivative must be at x = 0. So you get 2xN additional equations.
 

1. What is the FDM Heat Equation FTCS Scheme with NBC?

The FDM Heat Equation FTCS Scheme with NBC is a numerical method used to solve the heat equation, which describes the transfer of heat in a medium. It uses a finite difference method (FDM) to approximate the partial differential equations involved and the forward-time central-space (FTCS) scheme to discretize the equations in time and space. The Neumann boundary condition (NBC) specifies the heat flux at the boundary of the medium.

2. How does the FTCS scheme work?

The FTCS scheme uses a forward difference approximation for the time derivative and a central difference approximation for the space derivative. This means that the solution at each time step is calculated based on the values at the previous time step and the values at neighboring grid points. The scheme is first-order accurate in time and second-order accurate in space.

3. What is the significance of using the NBC in the FDM Heat Equation FTCS Scheme?

The NBC allows for a more accurate and stable numerical solution by specifying the heat flux at the boundary, rather than the temperature. This is especially important for problems with non-uniform boundary conditions, as it ensures that the heat flux is consistent across the boundary. It also prevents the solution from becoming unstable or oscillatory near the boundary.

4. What are the limitations of the FDM Heat Equation FTCS Scheme with NBC?

One limitation of this scheme is that it is only applicable to problems with constant or uniform boundary conditions. It also requires a relatively small time step for stability, which can make it computationally expensive for large systems. Additionally, the solution may become unstable if the time step is too large or the grid spacing is too coarse.

5. How does the FDM Heat Equation FTCS Scheme with NBC compare to other numerical methods?

The FDM Heat Equation FTCS Scheme with NBC is a simple and commonly used method for solving the heat equation. It is relatively easy to implement and can handle a wide range of boundary conditions. However, it is a first-order accurate method and can suffer from stability issues. Other numerical methods, such as the Crank-Nicolson scheme, may be more accurate and stable, but can be more complex to implement.

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