scalation.dynamics_pde

ParabolicPDE

Related Doc: package dynamics_pde

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 endpoints 0 and xm

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  1. final def !=(arg0: Any): Boolean

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

    Show the flaw by printing the error message.

    Show the flaw by printing the error message.

    method

    the method where the error occurred

    message

    the error message

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  14. final def notify(): Unit

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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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