A one-dimensional domain is discretized into $N$ sub-domains of width $\Delta x$ with node numbers $i=0,1,2,3, \ldots, N .$ If the time scale is discretized in steps of $\Delta t,$ the forward-time and centeredspace finite difference approximation at $i^{t h}$ node and $n^{t h}$ time step, for the partial differential equation $\frac{\partial v}{\partial t}=\beta \frac{\partial^{2} v}{\partial x^{2}}$ is
(A) $\frac{v_{i}^{(n+1)}-v_{i}^{(n)}}{\Delta t}=\beta\left[\frac{v_{i+1}^{(n)}-2 v_{i}^{(n)}+v_{i-1}^{(n)}}{(\Delta x)^{2}}\right]$
(B) $\frac{v_{i+1}^{(n+1)}-v_{i}^{(n)}}{\Delta t}=\beta\left[\frac{v_{i+1}^{(n)}-2 v_{i}^{(n)}+v_{i-1}^{(n)}}{2 \Delta x}\right]$
(C) $\frac{v_{i}^{(n)}-v_{i}^{(n-1)}}{\Delta t}=\beta\left[\frac{v_{i+1}^{(n)}-2 v_{i}^{(n)}+v_{i-1}^{(n)}}{(\Delta x)^{2}}\right]$
(D) $\frac{v_{i}^{(n)}-v_{i}^{(n-1)}}{2 \Delta t}=\beta\left[\frac{v_{i+1}^{(n)}-2 v_{i}^{(n)}+v_{i-1}^{(n)}}{2 \Delta x}\right]$

Question ID - 156996 | SaraNextGen Top Answer

A one-dimensional domain is discretized into $N$ sub-domains of width $\Delta x$ with node numbers $i=0,1,2,3, \ldots, N .$ If the time scale is discretized in steps of $\Delta t,$ the forward-time and centeredspace finite difference approximation at $i^{t h}$ node and $n^{t h}$ time step, for the partial differential equation $\frac{\partial v}{\partial t}=\beta \frac{\partial^{2} v}{\partial x^{2}}$ is
(A) $\frac{v_{i}^{(n+1)}-v_{i}^{(n)}}{\Delta t}=\beta\left[\frac{v_{i+1}^{(n)}-2 v_{i}^{(n)}+v_{i-1}^{(n)}}{(\Delta x)^{2}}\right]$
(B) $\frac{v_{i+1}^{(n+1)}-v_{i}^{(n)}}{\Delta t}=\beta\left[\frac{v_{i+1}^{(n)}-2 v_{i}^{(n)}+v_{i-1}^{(n)}}{2 \Delta x}\right]$
(C) $\frac{v_{i}^{(n)}-v_{i}^{(n-1)}}{\Delta t}=\beta\left[\frac{v_{i+1}^{(n)}-2 v_{i}^{(n)}+v_{i-1}^{(n)}}{(\Delta x)^{2}}\right]$
(D) $\frac{v_{i}^{(n)}-v_{i}^{(n-1)}}{2 \Delta t}=\beta\left[\frac{v_{i+1}^{(n)}-2 v_{i}^{(n)}+v_{i-1}^{(n)}}{2 \Delta x}\right]$