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Modeling Process
Brief overvew of the Virtual Cell modeling scheme.

Message Architecture
Explanation of the Virtual Cell messaging scheme.

Modeling Framework
Provides the biological abstractions necessary to model cellular physiology.

Math Framework
Provides general purpose solver for mathematical problems.

User Interface
Java based biological interface.

Testing Framework
Numerics and Expression Testing, Math Verifier.

Virtual Cell Grammar
Formal definition of language in BNF grammar.


Math Framework Validation Test Procedure

Purpose

This is the Validation Test Procedure for the Mathematics Framework of the Virtual Cell Project.  This procedure describes specific testing performed on the Mathematics Framework to determine the correctness of the code and the accuracy of various algorithms.  Questions or comments may be directed to schaff@neuron.uchc.edu.

Solution to Reaction Diffusion Equations

The numerical methods for the reaction diffusion equation must be validated for appropriate combinations of one, two or three dimensions Dirichlet or Neuman boundary conditions with or without membranes steady and unsteady solutions exact and calculated test functions finite difference and finite element

These solution must be validated using both remainders (artificially generated exact solutions) and, when possible, exact solutions.

Test #
dim
Boundary
membranes
reactions
state
solution
mesh
Test01
1
Dirichlet
no
1st Order
unsteady
remainders
FD
Test02
1
Dirichlet
no
no
unsteady
exact
FD
Test03
1
Dirichlet
no
1st Order
unsteady
exact
FD
Test04
2
Dirichlet
no
1st Order
unsteady
exact
FD
Test05
2
Dirichlet
no
no
unsteady
exact
FD
Test06
2
Dirichlet
no
1st Order
steady
exact
FD
Test07
3
Dirichlet
no
no
unsteady
exact
FD
Test08
3
Dirichlet
no
1st Order
unsteady
exact
FD
Test09
3
Dirichlet
no
1st Order
steady
exact
FD
Test10
3
Dirichlet
no
1st Order
unsteady
remainders
FD
Test11
2
Dirichlet
no
1st Order
unsteady
remainders
FD
Test12
1
Neumann
no
no
unsteady
exact
FD
Test13
2
Neumann
no
1st Order
unsteady
exact
FD
Test14
2
Neumann
no
1st Order
unsteady
remainders
FD
Test15
2
Neumann
no
1st Order
steady
exact
FD
Test16*
3
Neumann
no
1st Order
unsteady
exact
FD
Test17*
3
Neumann
no
1st Order
unsteady
remainders
FD
Test18*
3
Neumann
no
1st Order
steady
exact
FD
Test19
1
Dirichlet
1
no
unsteady
exact
FD
Test20
1
Dirichlet
1
1st Order
steady
exact
FD
Test21
1
Dirichlet
1
1st Order
unsteady
remainder
FD
Test22
2
Dirichlet
box in box
1st Order
unsteady
remainder
FD
Test23
1
Dirichlet
1(with membrane

species)
1st Order
unsteady
remainder
FD
Test24
1
Dirichlet
1
1st Order
unsteady
exact
FD

Test01 - 1 dimension, Dirichlet, no membranes, unsteady, remainders, FD

  • Equations:


    dC1/dt = D d2 C1/ dx2 - k1 C1 + k2 C2 + Remainder1(x,t)

    dC2/dt = D d2 C2/ dx2 + k1 C1 - k2 C2 + Remainder2(x,t)

    where D = 40 um2/sec,   k1 = 10 sec-1,   k2 = 50 sec-1

  • Test Functions:


    C1(x,t) = e-t (1 + sin(x + 1))

    C2(x,t) = e-t (1 + cos(x + 1))

  • Test Domain:


    x : [0, 10],    t : [0,1]

  • Test Results:
   time step 
   num x     L2 error(C1)     L2 error(C2) 
0.01 
10 
0.08266 
0.0390591 
0.001 
40 
0.00460041 
0.00234898 
0.0001 
160 
0.000308151 
0.000158134 
0.00001 
640 
2.1985e-5 
1.12512e-5 

Test02 - 1 dimension, Dirichlet, no membranes, unsteady, exact solution, FD

  • Equations:


    dC1/dt = D d2 C1/ dx2

    where D = 40 um/sec

  • Test Functions:


    Exact solution: C1(x,t) = e-t sin ( x D1-0.5 )

  • Test Domain:


    x : [0, 10],    t : [0,1]

  • Test Results:
   time step 
   num x 
  L2 error(C1) 
0.01 
10 
0.00197603 
0.001 
40 
0.000172504 
0.0001 
160 
1.592e-5 
0.00001 
640 
1.50969e-6 
0.00001
4
0.00560882
0.00001
8
0.0010971
0.00001
16
0.000247447
0.00001
32
5.98753e-5
0.00001
64
1.56415e-5

Test03 - 1 dimension, Dirichlet, no membranes, unsteady, exact solutions, FD

  • Equations:


    dC1/dt = D d2 C1/ dx2 - k1 C1 + k2 C2

    dC2/dt = D d2 C2/ dx2 + k1 C1 - k2 C2

    where D = 40 um2/sec,   k1 = 10 sec-1,   k2 = 50 sec-1

  • Test Functions:


    C1(x,t) = (1/K) e-2Kt [ k2 sin((2K/D)0.5  x) + k1 sin((K/D)0.5  x) ]

    C2(x,t) = (k1/K) e-2Kt [ sin((2K/D)0.5  x) - sin((K/D)0.5  x) ]

    where K = k1 + k2

  • Test Domain:


    x : [0, a],    t : [0,1]

    where a < 0.5 pi sqrt(D/2K)

  • Test Results:


    x : [0, 0.5],    t : [0, 0.01]
     

       time step 
    		    num x     L2 error(C1) 
       L2 error(C2) 
    0.001 
    4
    0.00333007
    0.0147477
    0.0001
    8
    0.000388162
    0.00163811
    0.00001
    16
    4.91031e-5
    0.000199155
    0.000001
    32
    7.1985e-6
    2.78007e-5
    0.0000001
    64
    1.27442e-6
    6.68866e-6

Test04 - 2 dimensions, Dirichlet, no membranes, unsteady, exact solution, FD

  • Equations:


    dC1/dt = D [ d2 C1/ dx2 + d2 C1/ dy2 ] - k1 C1 + k2 C2

    dC2/dt = D [ d2 C2/ dx2 + d2 C2/ dy2 ] + k1 C1 - k2 C2

    where D = 40 um2/sec,   k1 = 10 sec-1,   k2 = 50 sec-1

  • Test Functions: (Exact Solution)


    C1(x,t) = (1/K) e-2Kt [ k2 sin((K/D)0.5  (x + y) ) + k1 sin((K/2D)0.5  (x + y) ) ]

    C2(x,t) = (k1/K) e-2Kt [ sin((K/D)0.5  (x + y) ) - sin((K/2D)0.5  (x + y) ) ]

    where K = k1 + k2

  • Test Domain:


    x : [0, a],   y : [0, b],  t : [0, 0.01]

    where  a = b = 0.5    (note: a + b < 0.5 pi sqrt(D/K) )

  • Test Results:


    t : [0, 0.01]
     

    time step
    num x
    num y
    L2 error(C1)
    L2 error(C2)
    0.001
    4
    4
    0.0011852
    0.00560584
    0.0001
    8
    8
    0.000241767
    0.000784723
    0.00001
    16
    16
    2.81032e-5
    9.15467e-5
    0.000001
    32
    32
    ?
    ?

Test05 - 2 dimensions, Dirichlet, no membranes, unsteady, exact solution, FD

  • Equations:


    dC1/dt = D [ d2 C1/ dx2 + d2 C1/ dy2 ]

      where D = 40 um2/sec,

  • Test Functions: (Exact Solution)


    C1(x,t) = e-t   sin( (2D)-0.5  (x + y) )

  • Test Domain:


    x : [0, a],   y : [0, b],  t : [0,1]

    where  a = b =10    (note: a + b < pi sqrt(2D) )

  • Test Results:

     
     
     

      time step 
       num x
       num y
       L2 error(C1) 
    0.01
    10
    10
    ??
    0.001
    20
    20
    9.05786e-5
    0.0001
    40
    40
    1.38682e-5
    0.00001
    80
    80
    2.5669e-6

Test06 - 2 dimensions, Dirichlet, no membranes, steady state, exact solution, FD

  • Equations:


    dC1/dt = D [ d2 C1/ dx2 + d2 C1/ dy2 ] - k1 C1 + k2 C2

    dC2/dt = D [ d2 C2/ dx2 + d2 C2/ dy2 ] + k1 C1 - k2 C2

    where D = 40 um2/sec,   k1 = 10 sec-1,   k2 = 50 sec-1

  • Test Functions: (Exact Solution)


    C1(x,t) = (k2/K) ePy [ sin(P x) - sin( (P2 - K/D)0.5 x) ]

    C2(x,t) = (1/K) ePy [ k1 sin(P x)+ k2 sin( (P2 - K/D)0.5 x) ]

    where K = k1 + k2 ,     P = pi/(2 a)

  • Test Domain:


    x : [0, a],   y : [0, b],  t : [0,inf]

    where  a = 1, b = 0.5    (note: a < 0.5 pi sqrt(D/K),   b = a/2)

  • Test Results:


    time step = 1e-5 sec,   tolerance = 1e-10,
     

      num x
       num y
       L2 error(C1) 
       L2 error(C2) 
    		 end time matrix iterations
    20
    20
    7.32543e-5
    1.15737e-5
    		 0.01013
    2
    40
    40
    1.77765e-5
    2.84278e-6
    0.01081
    2
    80
    80
    4.39095e-6
    7.06993e-7
    0.01682
    2
    80
    80
    4.38755e-6
    7.06239e-7
    0.01358
    3

Test07 - 3 dimensions, Dirichlet, no membranes, unsteady, exact solution, FD

  • Equations:


    dC1/dt = D [ d2 C1/ dx2 + d2 C1/ dy2 + d2 C1/ dz2 ]

    where D = 40 um2/sec,

  • Test Functions: (Exact Solution)


    C1(x,t) = e-t   sin( (3D)-0.5  (x + y + z) )

  • Test Domain:


    x : [0, a],   y : [0, b],   z : [0, c],  t : [0,1]

    where  a = b = c = 5    (note: a + b + c < 0.5 pi sqrt(3D))

  • Test Results:


    x: [0,1],    y : [0, 1],    z : [0, 1],    t : [0, 1]
     

    time step
    num x
    num y
    num z
    L2 error(C1)
    matrix iterations
    0.001
    4
    4
    4
    6.24844e-6
    1
    0.0001
    8
    8
    8
    9.5342e-6
    1
    0.001
    4
    4
    4
    1.7897e-7
    2
    0.0001
    8
    8
    8
    1.56886e-7
    2
    0.001
    4
    4
    4
    1.75313e-7
    3
    0.0001
    8
    8
    8
    3.12236e-8
    3

     x: [0,5],    y : [0, 5],    z : [0, 5],    t : [0, 1]
     

    time step
    num x
    num y
    num z
    L2 error(C1)
    matrix iterations
    0.001
    4
    4
    4
    1.99009e-5
    1
    0.0001
    8
    8
    8
    5.02036e-6
    1
    0.001
    4
    4
    4
    1.98598e-5
    2
    0.0001
    8
    8
    8
    4.97963e-6
    2
    0.001
    4
    4
    4
    1.98598e-5
    3
    0.0001
    8
    8
    8
    4.97963e-6
    3

Test08 - 3 dimensions, Dirichlet, no membranes, unsteady, exact solution, FD

  • Equations:
    dC1/dt = D [ d2 C1/ dx2 + d2 C1/ dy2 + d2 C1/ dz2 ] - k1 C1 + k2 C2

    dC1/dt = D [ d2 C1/ dx2 + d2 C1/ dy2 + d2 C1/ dz2 ] + k1 C1 - k2 C2

    where D = 40 um2/sec,   k1 = 10 sec-1,   k2 = 50 sec-1

  • Test Functions: (Exact Solution)


    C1(x,t) = (1/K) e-2Kt [ k2 sin((2K/3D)0.5  (x + y + z) ) + k1 sin((K/3D)0.5  (x + y + z) ) ]

    C2(x,t) = (k1/K) e-2Kt [ sin((2K/3D)0.5  (x + y + z) ) - sin((K/3D)0.5  (x + y + z) ) ]

    where K = k1 + k2

  • Test Domain:


    x : [0, a],   y : [0, b],   z : [0, c],  t : [0, 0.01]

    where  a = b = c = 0.5    ( note: a + b + c < 0.5 pi sqrt(3D/2K) )

  • Test Results:


    x: [0,0.5,    y : [0, 0.5],    z : [0, 0.5],    t : [0, 0.01]
     

    time step
    num x
    num y
    num z
    L2 error(C1)
    L2 error(C2)
    matrix iterations
    0.0001
    4
    4
    4
    7.27345e-5
    0.000338918
    4
    0.00001
    8
    8
    8
    1.34937e-5
    5.94177e-5
    4
    0.000001
    16
    16
    16
    4

Test09 -3 dimensions, Dirichlet, no membranes, steady state, exact solution, FD

  • Equations:


    dC1/dt = D [ d2 C1/ dx2 + d2 C1/ dy2 + d2 C1/ dz2] - k1 C1 + k2 C2

    dC2/dt = D [ d2 C2/ dx2 + d2 C2/ dy2+ d2 C2/ dz2] ] + k1 C1 - k2 C2

    where D = 40 um2/sec,   k1 = 10 sec-1,   k2 = 50 sec-1

  • Test Functions: (Exact Solution)


    C1(x,t) = (k2/K)( ePy + ePz )[ sin(P x) - sin( (P2 - K/D)0.5 x) ]

    C2(x,t) = (1/K)( ePy+ ePz )[ k1 sin(P x)+ k2 sin( (P2 - K/D)0.5 x) ]

    where K = k1 + k2 ,     P = pi/(2 a)

  • Test Domain:


    x : [0, a],   y : [0, b],  y : [0, c], t : [0,inf]

    where  a = 1, b = c= 0.5    (note: a < 0.5 pi sqrt(D/K),   b = a/2)

  • Test Results:


    time step = 1e-5 sec,   tolerance = 1e-10,
     

      num x
       num y
    num z
       L2 error(C1) 
       L2 error(C2) 
    		 end time matrix iterations
    4
    4
    4
    0.000892889
    0.000124669
    		 0.00597
    4
    8
    8
    8
    0.000236968
    3.64568e-5
    0.00575
    4
    16
    16
    16
    5.78812e-5
    9.22472e-6
    0.00571
    4
    32
    32
    32
    1.42039e-5
    2.29892e-6
    0.00575
    4

Test10 - 3 dimensions, Dirichlet, no membranes, unsteady, remainders, FD

  • Equations:


    dC1/dt = D [ d2 C1/ dx2 + d2 C1/ dy2 + d2 C1/ dz2 ] - k1 C1 + k2 C2 + Remainder1(x,y,z,t)

    dC2/dt = D [ d2 C2/ dx2 + d2 C2/ dy2 + d2 C2/ dz2 ] + k1 C1 - k2 C2 + Remainder2(x,y,z,t)

    where D = 40 um2/sec,   k1 = 10 sec-1,   k2 = 50 sec-1

  • Test Functions:


    C1(x,y,z,t) = (1 + e-t )(1 + sin(x + y + z))

    C2(x,y,z,t) = (2 + e-t )(1 + cos(x + y + z))

  • Test Domain:


    x : [0, a],   y : [0, b],   z : [0, c],    t : [0, T]

    where a,b,c,T are arbitrary positive numbers

  • Test Results:


    x : [0, 1],   y: [0, 1],   z: [0, 1],    t : [0, 1]
     

    time step
    num x
    num y
    num z
    L2 error(C1)
    L2 error(C2)
    matrix iterations
    0.01 
    4
    4
    4
    0.000244817
    6.30903e-5
    4
    0.001 
    8
    8
    8
    5.92688e-5
    1.99093e-5
    4
    0.0001 
    16
    16
    16
    1.37372e-5
    4.89236e-6
    4
    0.00001 
    32
    32
    32
    ?? too long ??
    ?? too long ??
    4

Test11- 2 dimensions, Dirichlet, no membranes, unsteady, remainders, FD

  • Equations:


    dC1/dt = D [ d2 C1/ dx2 + d2 C1/ dy2 ] - k1 C1 + k2 C2 + Remainder1(x,y,t)

    dC2/dt = D [ d2 C2/ dx2 + d2 C2/ dy2 ] + k1 C1 - k2 C2 + Remainder2(x,y,t)

    where D = 40 um2/sec,   k1 = 10 sec-1,   k2 = 50 sec-1

  • Test Functions:


    C1(x,y,t) = (1 + e-t )(1 + sin(x + y ))

    C2(x,y,t) = (2 + e-t )(1 + cos(x + y ))

  • Test Domain:


    x : [0, a],    y: [0, b],    t : [0, T]

    where a,b,c,T are arbitrary positive numbers

  • Test Results:


    x : [0, 1],  y : [0, 1],   t : [0, 1]

time step
num x
num y
L2 error(C1)
L2 error(C2)
matrix iterations
0.01 
4
4
0.000325152
0.000198966
3
0.001 
8
8