In[]:=
Needs["NDSolve`FEM`"]​​γ=1.66;​​(*Ω=Rectangle[{0,0},{1,1}];*)​​Ω=Polygon[{{1,0},{.1,1},{0,1},{0,0}}]​​​​bcs={DirichletCondition[u[x,t]0,0≤x≤1&&0<y<1]};​​​​p[x_,t_]:=(γ-1)ρ[x,t](e[x,t]-u[x,t]^2/2);​​​​mesh=ToElementMesh[Ω,MaxCellMeasure0.01];​​mesh["Wireframe"]​​​​{uf,ρf,ef}=NDSolveValue​​
∂
t
ρ[x,t]+
∂
x
(ρ[x,t]*u[x,t])0,​​
∂
t
(ρ[x,t]*u[x,t])+
∂
x
(ρ[x,t]*
2
u[x,t]
+p[x,t])0,​​
∂
t
e[x,t]+
∂
x
(u[x,t]*(e[x,t]+p[x,t]))0,​​bcs,e[x,0]
101000
γ-1
,ρ[x,0]10,u[x,0]0​​,{u,ρ,e},{x,t}∈mesh,Method{"FiniteElement","InterpolationOrder"{u2,ρ2,e2}};​​​​ContourPlot[uf[x,y],{x,y}∈Ω,AspectRatioAutomatic,ColorFunction"Rainbow",Contours25]
Out[]=
Polygon
Number of points: 4
Embedding dimension: 2

Out[]=
FindRoot
:Linear equation encountered that has no solution.
FindRoot
:The step size in the search has become less than the tolerance prescribed by the PrecisionGoal option, but the function value is still greater than the tolerance prescribed by the AccuracyGoal option.
Out[]=
​