Question

A one-dimensional domain is discretized into $N$ subdomains of width $\Delta x$ with node numbers $i=0,1,2,3,...,N$. If the time scale is discretized in steps of $\Delta t,$ the forward-time and centered-space finite difference approximation at $i^{th}$ node and $n^{th}$ time step, for the partial differential equation $\frac{\partial v}{\partial t}=\beta \frac{{\partial }^{2}v}{\partial x^2}$ is

• A. $\frac{v_i^{\left(n\right)}-v_i^{(n-1)}}{2\Delta t}=\beta \left[\frac{v_{i+1}^{\left(n\right)}-2v_i^n+v_{i-1}^n}{2\Delta x}\right]$
• B. $\frac{v_i^{\left(n\right)}-v_i^{(n-1)}}{\Delta t}=\beta \left[\frac{v_{i+1}^{\left(n\right)}-2v_i^n+v_{i-1}^n}{(\Delta x{)}^2}\right]$
• C. $\frac{v_{i+1}^{(n+1)}-v_i^{\left(n\right)}}{\Delta t}=\beta \left[\frac{v_{i+1}^{\left(n\right)}-2v_i^n+v_{i-1}^n}{2\Delta x}\right]$
• D. $\frac{v_i^{(n+1)}-v_i^{\left(n\right)}}{\Delta t}=\beta \left[\frac{v_{i+1}^{\left(n\right)}-2v_i^n+v_{i-1}^n}{(\Delta x{)}^2}\right]$

Answer 1

The correct option is D $\frac{v_i^{(n+1)}-v_i^{\left(n\right)}}{\Delta t}=\beta \left[\frac{v_{i+1}^{\left(n\right)}-2v_i^n+v_{i-1}^n}{(\Delta x{)}^2}\right]$

Given differential equation
$\frac{\partial v}{\partial t}=\beta \frac{{\partial }^{2}v}{\partial x^2}$
Using forward time finte difference
$\frac{\partial v}{\partial t}=\frac{v_i^{(n+1)}-v_i^{\left(n\right)}}{\left(\Delta t\right)}$
Using centred space finite difference
$\frac{{\partial }^{2}v}{\partial x^2}=\frac{V_{i+1}^{\left(n\right)}-2v_i^{\left(n\right)}+v_{i-1}^{\left(n\right)}}{(\Delta x{)}^2}$
$[\text{as},\frac{{\partial }^{2}v}{\partial x^2}=\frac{f(x+h)-2f\left(x\right)+f(x-h)}{h^2}]$
Putting values in equation
$\frac{v_i^{(n+1)}-v_i^{\left(n\right)}}{\Delta t}=\beta \left[\frac{v_{i+1}^{\left(n\right)}-2v_i^n+v_{i-1}^n}{(\Delta x{)}^2}\right]$