Question

For the following 1D transient heat transfer PDE problem, please write the discretized

equations using finite difference method, where T[infinity] is the ambient temperature.
K ∂2T/∂x2=∂t/∂T0≤x≤L,t>0
k∂T/∂x =−ha(T−T[infinity])
at x=0,t>0
k∂T/∂x =0
at x=L,t>0
T==G(x)
t=0,0≤x≤L
(a) Using forward (explicit) method for time and forward method for x, forward or
backward difference methods for boundary conditions.
(b) Using the backward (implicit) method for time and central method for x and
boundary conditions.

Answer 1

To discretize the given 1D transient heat transfer PDE problem using the finite difference method:

(a) Using the forward (explicit) method for time and forward method for x, and forward or backward difference methods for boundary conditions:

• Discretize the heat equation using the forward difference method for time and forward method for x.
• Discretize the boundary conditions using either the forward or backward difference method.

(b) Using the backward (implicit) method for time and central method for x and boundary conditions:

• Discretize the heat equation using the backward difference method for time and central method for x.
• Discretize the boundary conditions using the central difference method.

Step-by-step explanation:

To discretize the given 1D transient heat transfer PDE problem using the finite difference method, there are two main approaches you can take. Let's go through each one:

1. Forward (explicit) method for time and forward method for x:

• - Discretize the heat equation using the forward difference method for time and forward method for x. This involves approximating the time derivative using a forward difference scheme and the spatial derivative using a forward difference scheme.
• - Discretize the boundary conditions using either the forward or backward difference method. For example, if the boundary conditions are given at both ends of the 1D domain, you can use a forward difference scheme at one boundary and a backward difference scheme at the other.

2. Backward (implicit) method for time and central method for x and boundary conditions:

• - Discretize the heat equation using the backward difference method for time and central difference method for x. This involves approximating the time derivative using a backward difference scheme and the spatial derivative using a central difference scheme.
• - Discretize the boundary conditions using the central difference method. This means approximating the boundary conditions using a central difference scheme.

Both approaches have their own advantages and disadvantages. The forward method for time is easy to implement and computationally efficient, but it may introduce stability issues and require smaller time steps for accuracy. On the other hand, the backward method for time is unconditionally stable but involves solving a system of equations at each time step, making it computationally more expensive.