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In [2]:



    


using ApproxFun,Interact,Reactive

Advection-diffusion with Dirichlet $$u_t = 0.01 \Delta u -4 u_x -3 u_y$$ $$u(\pm 1,y,t)=u(x,\pm 1,t)=0$$





In [3]:



    


d=Interval()^2
u0   = ProductFun((x,y)->exp(-40(x-.1)^2-40(y+.2)^2),d)
B=dirichlet(d);D=Derivative(Interval())
L=(0.01D^2-4D)⊗I + I⊗(0.01D^2-3D)

glp=Input(u0);lift(f->ApproxFun.contour(f;levels=0.001:.1:1.),glp)



    


















    
Out[3]:









  
    x
  
  
    -3.5
    -3.0
    -2.5
    -2.0
    -1.5
    -1.0
    -0.5
    0.0
    0.5
    1.0
    1.5
    2.0
    2.5
    3.0
    3.5
    -3.0
    -2.9
    -2.8
    -2.7
    -2.6
    -2.5
    -2.4
    -2.3
    -2.2
    -2.1
    -2.0
    -1.9
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    -1.6
    -1.5
    -1.4
    -1.3
    -1.2
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    -1.0
    -0.9
    -0.8
    -0.7
    -0.6
    -0.5
    -0.4
    -0.3
    -0.2
    -0.1
    0.0
    0.1
    0.2
    0.3
    0.4
    0.5
    0.6
    0.7
    0.8
    0.9
    1.0
    1.1
    1.2
    1.3
    1.4
    1.5
    1.6
    1.7
    1.8
    1.9
    2.0
    2.1
    2.2
    2.3
    2.4
    2.5
    2.6
    2.7
    2.8
    2.9
    3.0
    -4
    -2
    0
    2
    4
    -3.0
    -2.8
    -2.6
    -2.4
    -2.2
    -2.0
    -1.8
    -1.6
    -1.4
    -1.2
    -1.0
    -0.8
    -0.6
    -0.4
    -0.2
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    1.2
    1.4
    1.6
    1.8
    2.0
    2.2
    2.4
    2.6
    2.8
    3.0
  
  
    
      1.00
      0.25
      0.75
      0.00
      0.50
    
    
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
        
        
        
        
        
      
    
    
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    -3.5
    -3.0
    -2.5
    -2.0
    -1.5
    -1.0
    -0.5
    0.0
    0.5
    1.0
    1.5
    2.0
    2.5
    3.0
    3.5
    -3.0
    -2.9
    -2.8
    -2.7
    -2.6
    -2.5
    -2.4
    -2.3
    -2.2
    -2.1
    -2.0
    -1.9
    -1.8
    -1.7
    -1.6
    -1.5
    -1.4
    -1.3
    -1.2
    -1.1
    -1.0
    -0.9
    -0.8
    -0.7
    -0.6
    -0.5
    -0.4
    -0.3
    -0.2
    -0.1
    0.0
    0.1
    0.2
    0.3
    0.4
    0.5
    0.6
    0.7
    0.8
    0.9
    1.0
    1.1
    1.2
    1.3
    1.4
    1.5
    1.6
    1.7
    1.8
    1.9
    2.0
    2.1
    2.2
    2.3
    2.4
    2.5
    2.6
    2.7
    2.8
    2.9
    3.0
    -4
    -2
    0
    2
    4
    -3.0
    -2.8
    -2.6
    -2.4
    -2.2
    -2.0
    -1.8
    -1.6
    -1.4
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    -0.6
    -0.4
    -0.2
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    1.2
    1.4
    1.6
    1.8
    2.0
    2.2
    2.4
    2.6
    2.8
    3.0
  
  
    y
  



  





















In [4]:



    


u0=BDF4(B,L,u0,0.002,150,glp);

Wave equation with Dirichlet $$u_{tt} = \Delta u$$ $$u(x,y,0)=u_0(x,y), u(\pm 1,y,t)=u(x,\pm 1,t)=0$$





In [5]:



    


d=Interval()^2
# initial condition
u0   = ProductFun((x,y)->exp(-50x^2-50y^2),d)
B= dirichlet(d) ;L=lap(d);
glp=Input(u0);lift(f->ApproxFun.contour(f,levels=-0.30001:0.02:1.0),glp)



    


















    
Out[5]:









  
    x
  
  
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    -3.0
    -2.5
    -2.0
    -1.5
    -1.0
    -0.5
    0.0
    0.5
    1.0
    1.5
    2.0
    2.5
    3.0
    3.5
    -3.0
    -2.9
    -2.8
    -2.7
    -2.6
    -2.5
    -2.4
    -2.3
    -2.2
    -2.1
    -2.0
    -1.9
    -1.8
    -1.7
    -1.6
    -1.5
    -1.4
    -1.3
    -1.2
    -1.1
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    -0.9
    -0.8
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    -0.6
    -0.5
    -0.4
    -0.3
    -0.2
    -0.1
    0.0
    0.1
    0.2
    0.3
    0.4
    0.5
    0.6
    0.7
    0.8
    0.9
    1.0
    1.1
    1.2
    1.3
    1.4
    1.5
    1.6
    1.7
    1.8
    1.9
    2.0
    2.1
    2.2
    2.3
    2.4
    2.5
    2.6
    2.7
    2.8
    2.9
    3.0
    -4
    -2
    0
    2
    4
    -3.0
    -2.8
    -2.6
    -2.4
    -2.2
    -2.0
    -1.8
    -1.6
    -1.4
    -1.2
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    -0.8
    -0.6
    -0.4
    -0.2
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    1.2
    1.4
    1.6
    1.8
    2.0
    2.2
    2.4
    2.6
    2.8
    3.0
  
  
    
      1.0
      0.0
      0.5
    
    
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
        
        
        
      
    
    
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    -3.5
    -3.0
    -2.5
    -2.0
    -1.5
    -1.0
    -0.5
    0.0
    0.5
    1.0
    1.5
    2.0
    2.5
    3.0
    3.5
    -3.0
    -2.9
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    -2.6
    -2.5
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    0.0
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    0.2
    0.3
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    0.5
    0.6
    0.7
    0.8
    0.9
    1.0
    1.1
    1.2
    1.3
    1.4
    1.5
    1.6
    1.7
    1.8
    1.9
    2.0
    2.1
    2.2
    2.3
    2.4
    2.5
    2.6
    2.7
    2.8
    2.9
    3.0
    -4
    -2
    0
    2
    4
    -3.0
    -2.8
    -2.6
    -2.4
    -2.2
    -2.0
    -1.8
    -1.6
    -1.4
    -1.2
    -1.0
    -0.8
    -0.6
    -0.4
    -0.2
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    1.2
    1.4
    1.6
    1.8
    2.0
    2.2
    2.4
    2.6
    2.8
    3.0
  
  
    y
  



  





















In [6]:



    


u0=BDF22(B,L,u0,0.004,200,glp);

Sine Gordon $$u_{tt}=\Delta u - \sin(u)$$





In [7]:



    


d=Interval()^2
u0   = ProductFun((x,y)->exp(-50x^2-50y^2),d)
B= dirichlet(d);L=lap(d)-I;g(u)=u-sin(u)

glp=Input(u0)
lift(ApproxFun.contour,glp)



    


















    
Out[7]:









  
    x
  
  
    -3.5
    -3.0
    -2.5
    -2.0
    -1.5
    -1.0
    -0.5
    0.0
    0.5
    1.0
    1.5
    2.0
    2.5
    3.0
    3.5
    -3.0
    -2.9
    -2.8
    -2.7
    -2.6
    -2.5
    -2.4
    -2.3
    -2.2
    -2.1
    -2.0
    -1.9
    -1.8
    -1.7
    -1.6
    -1.5
    -1.4
    -1.3
    -1.2
    -1.1
    -1.0
    -0.9
    -0.8
    -0.7
    -0.6
    -0.5
    -0.4
    -0.3
    -0.2
    -0.1
    0.0
    0.1
    0.2
    0.3
    0.4
    0.5
    0.6
    0.7
    0.8
    0.9
    1.0
    1.1
    1.2
    1.3
    1.4
    1.5
    1.6
    1.7
    1.8
    1.9
    2.0
    2.1
    2.2
    2.3
    2.4
    2.5
    2.6
    2.7
    2.8
    2.9
    3.0
    -4
    -2
    0
    2
    4
    -3.0
    -2.8
    -2.6
    -2.4
    -2.2
    -2.0
    -1.8
    -1.6
    -1.4
    -1.2
    -1.0
    -0.8
    -0.6
    -0.4
    -0.2
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    1.2
    1.4
    1.6
    1.8
    2.0
    2.2
    2.4
    2.6
    2.8
    3.0
  
  
    
      0.6
      0.4
      1.0
      0.2
      0.0
      0.8
    
    
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
        
        
        
        
        
        
      
    
    
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    -3.5
    -3.0
    -2.5
    -2.0
    -1.5
    -1.0
    -0.5
    0.0
    0.5
    1.0
    1.5
    2.0
    2.5
    3.0
    3.5
    -3.0
    -2.9
    -2.8
    -2.7
    -2.6
    -2.5
    -2.4
    -2.3
    -2.2
    -2.1
    -2.0
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    -0.6
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    -0.3
    -0.2
    -0.1
    0.0
    0.1
    0.2
    0.3
    0.4
    0.5
    0.6
    0.7
    0.8
    0.9
    1.0
    1.1
    1.2
    1.3
    1.4
    1.5
    1.6
    1.7
    1.8
    1.9
    2.0
    2.1
    2.2
    2.3
    2.4
    2.5
    2.6
    2.7
    2.8
    2.9
    3.0
    -4
    -2
    0
    2
    4
    -3.0
    -2.8
    -2.6
    -2.4
    -2.2
    -2.0
    -1.8
    -1.6
    -1.4
    -1.2
    -1.0
    -0.8
    -0.6
    -0.4
    -0.2
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    1.2
    1.4
    1.6
    1.8
    2.0
    2.2
    2.4
    2.6
    2.8
    3.0
  
  
    y
  



  





















In [8]:



    


u0=BDF22(B,L,g,u0,0.004,300,glp);

Content source: yuyichao/ApproxFun.jl

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