ndsolve with multiple initial conditions

Question

I am trying to solve a PDE using the NDsolve with initial and boundary conditions,

NDSolve[{D[u[t, x], t] - D[ D[u[t, x], x]] + D[u[t, x], x] == 10,
    u[0, x] == 2,
    u[t, 0] == 1,
    u[t, 2] == 1
    },
    u,
    {t, 0, 5}, {x, 0, 2}, MaxStepSize -> 0.01]

it worked. But when I trying to assign a initial condition: u[0,x]=2 when 0.5<=x<=1, u[0,x]=1 elsewhere in [0,2] by means of If

NDSolve[{D[u[t, x], t] - D[ D[u[t, x], x]] + D[u[t, x], x] == 10,
    If[0.5 <= x <= 1, u[0, x] == 2, u[0, x] == 1],
    u[t, 0] == 1,
    u[t, 2] == 1
    },
    u,
    {t, 0, 5}, {x, 0, 2}, MaxStepSize -> 0.01]

the system returned me

NDSolve::deqn: Equation or list of equations expected instead of If[0.5<=x<=1,u[0,x]==2,u[0,x]==1] in the first argument {(u^(1,0))[t,x]==10,If[0.5<=x<=1,u[0,x]==2,u[0,x]==1],u[t,0]==1,u[t,2]==1}.

I know something is wrong with my initial conditions expression, But how I assign the initial conditions with an if condition. Or a better way to deal with it?

Thank you for your time!

UPDATE:

I followed the advice of @bbgodfrey by using DirichletCondition. it worked.

NDSolve[{D[u[t, x], t] - D[ D[u[t, x], x]] + D[u[t, x], x] == 0,
        DirichletCondition[u[t, x] == 2, 0.5 <= x <= 1],
        DirichletCondition[u[t, x] == 1, {2 >= x > 1, 0 <= x < 0.5}],
    u[t, 0] == 1,
    u[t, 2] == 1
    },
    u,
    {t, 0, 5}, {x, 0, 2}, MaxStepSize -> 0.01]

But comparing with the results of @kglr, the results are different. this is the results using DirichletCondition, which also with an alert

NDSolve::femcscd: The PDE is convection dominated and the result may not be stable. Adding artificial diffusion may help.

this is the result using Boole

Thank you!

Answer 1

I like UnitStep.

NDSolve[{D[u[t, x], t] - D[D[u[t, x], x]] + D[u[t, x], x] == 10, 
   u[0, x] == 1 + UnitStep[x - 0.5] - UnitStep[x - 1], u[t, 0] == 1, 
   u[t, 2] == 1}, u[t, x], {t, 0, 5}, {x, 0, 2}, 
  MaxStepSize -> 0.01] // Flatten

u[t_, x_] = u[t, x] /. %

Plot3D[u[t, x], {t, 0, 5}, {x, 0, 2}]

Plot[u[0, x], {x, 0, 2}, PlotRange -> All]

Piecewise probably works too.

Answer 2

The third opinion, a simple modification of the author's code immediately leads to a solution. Note the solution near t=0

s = NDSolve[{D[u[t, x], t] - D[D[u[t, x], x]] + D[u[t, x], x] == 10, 
   u[0, x] == If[0.5 <= x <= 1, 2, 1], u[t, 0] == 1, u[t, 2] == 1}, 
  u, {t, 0, 5}, {x, 0, 2}]

{Plot3D[u[t, x] /. s, {t, 0, 5}, {x, 0, 2}, PlotRange -> All, 
  Mesh -> None, ColorFunction -> Hue, AxesLabel -> {"t", "x", ""}], 
 Plot3D[u[t, x] /. s, {t, 0, 1}, {x, 0, 2}, PlotRange -> All, 
  Mesh -> None, ColorFunction -> Hue, AxesLabel -> {"t", "x", ""}], 
 Plot3D[u[t, x] /. s, {t, 0, .05}, {x, 0, 2}, PlotRange -> All, 
  Mesh -> None, ColorFunction -> Hue, AxesLabel -> {"t", "x", ""}], 
 Plot[Evaluate[Table[u[t, x] /. s, {t, 0, .1, .01}]], {x, 0, 2}, 
  AxesLabel -> {"x", "u"}]}

We check the solution at t->= 0, and see that everything is not smooth there. We need a special numerical method to solve this problem. To eliminate the oscillations at t-> 0, add the option

s = NDSolve[{D[u[t, x], t] - D[D[u[t, x], x]] + D[u[t, x], x] == 10, 
   u[0, x] == If[0.5 <= x <= 1, 2, 1], u[t, 0] == 1, u[t, 2] == 1}, 
  u, {t, 0, 5}, {x, 0, 2}, MaxStepSize -> 0.01]

Answer 3

sol = NDSolve[{D[u[t, x], t] - D[D[u[t, x], x]] + D[u[t, x], x] == 10,
      u[0, x] == 1 + Boole[0.5 <= x <= 1], u[t, 0] == 1, 
     u[t, 2] == 1}, u, {t, 0, 5}, {x, 0, 2}, MaxStepSize -> 0.01][[1]];

{u[0, .75], u[0, .1 .5]} /. sol

{2., 1.}

Plot3D[Evaluate[u[t, x] /. sol], {t, 0, 5}, {x, 0, 2}]