Packages

  • package root
    Definition Classes
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  • package scalation

    The scalation package specifies system-wide constants for directory paths.

    The scalation package specifies system-wide constants for directory paths. Sub-packages may wish to define 'BASE-DIR = DATA_DIR + ⁄ + <package>' in their own 'package.scala' files. For maintainability, directory paths should only be specified in 'package.scala' files.

    Definition Classes
    root
  • package dynamics_pde

    The dynamics_pde package contains classes, traits and objects for system dynamics simulations using Partial Differential Equations 'PDE's.

    The dynamics_pde package contains classes, traits and objects for system dynamics simulations using Partial Differential Equations 'PDE's.

    Definition Classes
    scalation
  • FirstOrderPDE
  • FirstOrderPDETest
  • FirstOrderPDETest2
  • FirstOrderPDETest3
  • ParabolicPDE
  • ParabolicPDETest

class ParabolicPDE extends Error

The ParabolicPDE' class is used to solve parabolic partial differential equations like the Heat Equation. Let 'u(x, t)' = temperature of a rod at position '0 <= x <= xm' and time 't' > 0. Numerically solve the

Heat Equation: u_t = k * u_xx with initial conditions u(x, 0) = ic(x) boundary conditions (u(0, t), u(xm, t)) = bc

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Instance Constructors

  1. new ParabolicPDE(k: Double, dt: Double, dx: Double, xm: Double, ic: FunctionS2S, bc: (Double, Double))

    k

    the thermal conductivity

    dt

    delta 't'

    dx

    delta 'x'

    xm

    the length of the rod

    ic

    the initial conditions as a function of position 'x'

    bc

    the boundary conditions as a 2-tuple for end-points 0 and 'xm'

Value Members

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  9. final def flaw(method: String, message: String): Unit

    Show the flaw by printing the error message.

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  16. def solve(t: Double): VectorD

    Solve for the temperature of the rod at time 't', returning the vector of temperatures representing the temperature profile of the rod over its length.

    Solve for the temperature of the rod at time 't', returning the vector of temperatures representing the temperature profile of the rod over its length. This method uses an explicit finite difference technique to solve the PDE.

    t

    the time the solution is desired

  17. def solveCN(t: Double): VectorD

    Solve for the temperature of the rod at time 't', returning the vector of temperatures representing the temperature profile of the rod over its length.

    Solve for the temperature of the rod at time 't', returning the vector of temperatures representing the temperature profile of the rod over its length. This method uses the implicit Crank-Nicolson technique to solve the PDE, which provides greater stability and accuracy. Implicit recurrence equation: -r*u(i-1, j2) + 2.*(1.+r)*u(i, j2) - r*u(i+1, j2) = r*u(i-1, j1) + 2.*(1.-r)*u(i, j1) + r*u(i+1, j1) This equation is solved simultaneously: solve for 'u' in 'mat * u = vec'

    t

    the time the solution is desired

    See also

    people.sc.fsu.edu/~jpeterson/5-CrankNicolson.pdf

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