Consider the initial solution of triangular shape

$u^{(0)}(x)=0$

$=10(x-0\mathrm{.}9)$

$=10(1\mathrm{.}1-x)$

$=0$

$x0\mathrm{.}9$

$0\mathrm{.}9x1\mathrm{.}0$

$1\mathrm{.}0x1\mathrm{.}1$

$x1\mathrm{.}1$

Compare with the exact solution

$u(x,t)=u^{(0)}(x-at)$

$a=1,x=0\mathrm{.}05$ and $41$ mesh points over the initial domain between $x=0$ and $x=2$

Use the following schemes

explicit central scheme $,u_i^{n+1}=u_i^n-\frac{at}{2x}(u_{i+1}^n-u_{i-1}^n),=a\frac{t}{}x$ equal to $0\mathrm{.}8$

The first order scheme

$\frac{u_i^{n+1}-u_i^n}{t}+\frac{a}{x}(u_i^n-u_{i-1}^n)=0$

$=a\frac{t}{}x$ equal to $0\mathrm{.}8$

$=a\frac{t}{}x$ equal to $1\mathrm{.}5$