Consider the PDE ut + Bux = a²uxx In the following, you do NOT have to demonstrate the accuracy of the schemes. At (4x)² and 12 in your schemes. = At Ax You may use notations ₁ a². (a) Write down an explicit scheme for approximating this PDE, which is first order accurate in time and second order accurate in space. (Your final scheme should be in the form of Ui,j+1 = ...) (b) Write down an implicit scheme for approximating this PDE, which is second order accurate in time and second order accurate in space and which only uses 2 time levels (that is this is still a one-time step method, not multi-time step method). (Hint: Refer to the derivation of Crank-Nicolson method in Sec 12.2)
Consider the PDE ut + Bux = a²uxx In the following, you do NOT have to demonstrate the accuracy of the schemes. At (4x)² and 12 in your schemes. = At Ax You may use notations ₁ a². (a) Write down an explicit scheme for approximating this PDE, which is first order accurate in time and second order accurate in space. (Your final scheme should be in the form of Ui,j+1 = ...) (b) Write down an implicit scheme for approximating this PDE, which is second order accurate in time and second order accurate in space and which only uses 2 time levels (that is this is still a one-time step method, not multi-time step method). (Hint: Refer to the derivation of Crank-Nicolson method in Sec 12.2)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider the PDE
ut + βux = aluxx
In the following, you do NOT have to demonstrate the accuracy of the schemes.
At
(Ax)²
your schemes.
(a) Write down an explicit scheme for approximating this PDE, which is first order accurate in time and second
order accurate in space. (Your final scheme should be in the form of Uį,j+1 = ···)
(b) Write down an implicit scheme for approximating this PDE, which is second order accurate in time and
second order accurate in space and which only uses 2 time levels (that is this is still a one-time step
method, not multi-time step method). (Hint: Refer to the derivation of Crank-Nicolson method in Sec 12.2)
At
You may use notations λι = α2. and 1₂ = ß
Batin
in
Ax
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