Clear[u,x,t];
{a,b}= {0, 12 };
TMax= 1.2;
uSol[x_,t_]=u[x,t]/.NDSolve[
{
D[ u[x,t],{t,1}] ==  D[ u[x,t], {x,2}]+0.8 u[x,t]D[u[x,t],x],
u[a,t]==0,
u[b,t]==0,
u[x,0]==(x-a)Sin[x](x-b)
},
u[x,t],
{x, a, b},
{t, 0, TMax},
Method->{"MethodOfLines", "SpatialDiscretization"->{"TensorProductGrid", "MinPoints"->300}}
][[1]]

Options[NDSolve]

?AccuracyGoal
?PrecisionGoal

ySol[t_]=y[t]/.NDSolve[
{
y''[t] == -y[t],
y'[0]==0.0,
y[0]==1.0
},
y[t],
{t, 0, 120}
][[1]]
ySol2[t_]=y[t]/.NDSolve[
{
y''[t] == -y[t],
y'[0]==0.0,
y[0]==1.0
},
y[t],
{t, 0, 120},
PrecisionGoal->2
][[1]]
Plot[{Cos[t],ySol[t],ySol2[t]},{t, 0, 120}]

Plot[{ySol''''[t],ySol2''''[t]},{t, 119, 120}]

?NDSolve

Clear[u,x,t];
{a,b}= {0, 12 };
TMax= 1.2;
uSol[x_,t_]=u[x,t]/.NDSolve[
{
D[ u[x,t],{t,1}] ==  D[ u[x,t], {x,2}]+0.8 u[x,t]D[u[x,t],x],
u[a,t]==0,
u[b,t]==0,
u[x,0]==(x-a)Sin[x](x-b)
},
u[x,t],
{x, a, b},
{t, 0, TMax}
][[1]]

Plot3D[ uSol[x,t],
{x, a, b},
{t, 0, TMax},
PlotRange->All
]