GATE Civil Engineering (CE) 2019 Shift 1 Solved Paper

Question : 65 of 65

Marks: +1, -0

A one-dimensional domain is discretized into

{N}

sub-domains of width

∆ {x}

with node numbers

{i}=0,1,2,3, . . ., {N}

. If the time scale is discretized in steps of

∆ {t}

, the forward-time and centered-space finite difference approximation at

{n}^{\;\text "th "\;}

node and nth time step, for the partial differential equation

\;{𝜕 v}/{d t}=β \;{𝜕x^{2} v}/{∂ x^{2}}

$\;{v_{i+1}^{(n+1)}-v_{i}^{(n)}}/{∆ t}=β[\;{v_{i+1}^{(n)}-2 v_{i}^{(n)}+v_{i-1}^{(n)}}/{2 ∆ x}]$
$\;{v_{i}^{(n)}-v_{i}^{(n-1)}}/{2 ∆ t}=β[\;{v_{i+1}^{(n)}-2 v_{i}^{(n)}+v_{i-1}^{(n)}}/{2 ∆ x}]$
$\;{ {v}_{ {i}}^{( {n}+1)}- {v}_{ {i}}^{( {n})}}/{∆ {t}}=β[\;{ {v}_{ {i}+1}^{( {n})}-2 {v}_{ {i}}^{( {n})}+ {v}_{ {i}-1}^{( {n})}}/{(∆ {x})^{2}}]$
$\;{v_{i}^{(n)}-v_{i}^{(n-1)}}/{∆ t}=β[\;{v_{i+1}^{(n)}-2 v_{i}^{(n)}+v_{i-1}^{(n)}}/{(∆ x)^{2}}]$

Solution:

Given differential equation

\;{∂ v}/{∂ t}=β \;{∂^{2} v}/{∂ x^{2}}

\;{∂ v}/{∂ t}=\;{v_{1}^{(n+1)}-v_{1}^{(n)}}/{(∆ t)}

.....(i)
Using forward time finite difference

\;{∂^{2} v}/{∂ x^{2}}=\;{v_{l+1}^{(j)}-2 v_{1}^{(n)}+v_{1-1}^{n}}/{(∆ x)^{2}}

...(ii)
Using centred space finite difference

\;{∂^{2} v}/{a x^{2}}=\;{f(x+h)-2 f(x)+f(x-h)}/{h^{2}}

Putting (i) and (ii) in PDE

\;{v_{1}^{(n+1)}-v_{1}^{(n)}}/{(∆ t)}=β(\;{v_{1+1}^{(p)}-2 v_{1}^{(n)}+v_{1-1}^{(n])}}/{∆ x^{2}})

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