Ordinary Differential Equations
Solve ordinary differential equations with finit differential method.
Problems
Initial value problem
HydrodynamicStability.ODEProblem — Type`Ordinary Differential Equations Problem`Define the ODEs problem
Problem type
ODEProblem{F, uType, yType, P, K} <: AbstractODEProblemFields
f: Function for the ordianry differential equations $du = f(u,t,p)$.u0: Initial conditions for the ODEs.yspan: Wall-normal direction span for the problem.p: The parameters for the problem. Defaults toNullParameterskwargs: The keyword arguments.
Boundary value problem
The general govening equation can be written as
\[ u' = f(u,t), \]
with the boundary conditions
\[ u(0) = a,\quad u(1) = b.\]
Following the mathematical defination, the numerical specification is given as
HydrodynamicStability.BVProblem — Type`Boundary Value Problem`Define a boundary Value problem
Problem type
BVProblem{F, BC, U0, Y, P, K} <: AbstractBVProblemFields
f: Function for the ordianry differential equations $du = f(u,t,p)$.bc: Boundary conditions for the ODEs. Given as a function.yspan: Wall-normal direction span for the problem.u0: Initial conditions for the ODEs.p: The parameters for the problem. Defaults toNullParameterskwargs: The keyword arguments.
Linear boundary value problem
For the linear problem, the equations can be reduced to first-order vector form, which take the matrix form
\[ A\boldsymbol{u(t)}' + D\boldsymbol{u(t)} = F(t). \]
Nonlinear boundary value problem
For the nonlinear problem, the equations can be reduced to first-order vector form, which take the matrix form
\[ A\boldsymbol{u}(t)' + D\boldsymbol{u}(t) = N(\boldsymbol{u},t) + F(t). \]
<!– Here, the superscript and subscript choose as a n*n Matrix with subscript i row, j column, and superscript k donate the iteration steps. –> Then, in order to use the New-Raphson method to solve the nonlinear problem $R(\boldsymbol{u}^k)=0$, the equations should be written as the Jacobian matrix that
\[ \boldsymbol{u}^{k+1} = \boldsymbol{u}^k-J^{-1}(\boldsymbol{u}^k)R(\boldsymbol{u}^k),\]
where
\[ R \begin{bmatrix}\boldsymbol{u}_1 \\ \boldsymbol{u}_2 \\ \vdots \\ \boldsymbol{u}_n \end{bmatrix} = M\boldsymbol{u}-\begin{bmatrix}N(\boldsymbol{u}_1,t_1) + F(t_1) \\ N(\boldsymbol{u}_2,t_2) + F(t_2) \\ \vdots \\ N(\boldsymbol{u}_n,t_n) + F(t_n)\end{bmatrix}, \]
with
\[ M = \mathrm{kron}(Diff,A)+\mathrm{kron}(I,D),\]
and
\[ J(\boldsymbol{u}) = M-\begin{bmatrix}N'(\boldsymbol{u}_1,t_1) & 0 & \cdots & 0 \\ 0 & N'(\boldsymbol{u}_2,t_2) & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & N'(\boldsymbol{u}_n,t_n) \end{bmatrix}.\]
Then, to deal with the boundary conditions,
The shooting method can be used to solve the nonlinear boundary value problem as well.
Solution
HydrodynamicStability.ODESolution — Type`ODE Solution`Definde the ordinary differential equations solution
Solution type
ODESolution{uType, yType, P, A, uType2, RT} <: AbstractODESolutionFields
u: The solution of the ODEs.y: The wall-normal direction span for the problem.prob: The problem that was solved.alg: The algorithm that was used to solve the problem.u_analytic: The analytic solution of the ODEs.retcode: The return code of the solver.
Algorithm
4th-order Runge-Kutta method
Shooting Method
Finite Diffenrence Method
Chebyshev collection
The discrete schemes
4-order center differential is used
\[ \frac{\mathrm{d}y_i}{\mathrm{d}t} = \frac{y_{i-2}-8y_{i-1}+8y_{i+1}-y_{i+2}}{12\Delta t}, \]
Example
Airy function
\[u'' - t u = 0,\]
with the boundary condition
\[u(0) = \operatorname{Ai}(0),\\ u(10) = \operatorname{Ai}(10),\]
where $\operatorname{Ai}(x)$ is the Airy function of the first kind.
The equation can be written as the first order form
\[ y_1' = y_2,\\ y_2' = ty_1.\]
Furthmore, write it in the matrix form
\[ \varGamma\frac{\mathrm{d} y_{i}}{\mathrm{d} t} + D y_{i} = 0,\]
where
\[ \varGamma=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad D=\begin{pmatrix} 0 & -1 \\ -t & 0 \end{pmatrix}.\]
Nonlinear boundary value problem
Mathematical Specification of the Nonlinear Problem
\[ u'' = u - u^2,\]
with the boundary condition
\[ u(0) = 1,\quad u(1) = 4.\]
Numerical Specification of the Nonlinear Problem
The equation can be write in the matrix form
\[ A\boldsymbol{u}(t)' + D\boldsymbol{u}(t) = N(\boldsymbol{u},t). \]
where the coefficient matrix are
\[ A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix},\quad D = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix},\]
and the nonlinear term is
\[ N(\boldsymbol{u},t) = \begin{bmatrix} 0 \\ - u^2 \end{bmatrix}, \quad DN(\boldsymbol{u},t) = \begin{bmatrix} 0 & 0 \\ -2u & 0 \end{bmatrix}.\]
Blasius equation
Mathematical Specification of the Blasius Equation
solve the Blasius equation
\[ f''' + \frac{1}{2} f f'' = 0,\]
with the boundary condition
\[ f(0) = f'(0) = 0, \quad f' \to 1 \quad \mathrm{as} \quad y \to \infty.\]
Numerical Specification of the Blasius Equation
Numerical algorithm: Shooting method
The equation can be write as the first order ODE system
du[1] = u[2]
du[2] = u[3]
du[3] = -1 / 2 * u[1] * u[3]with the boundary condition
residual[1] = u[begin][1]
residual[2] = u[begin][2]
residual[3] = u[end][2] - 1The domain is [0, 15], and the initial guess is [0, 0, 0.3].
Numerical algorithm: NSFDM
The equations can be write in the matrix form
\[ A\boldsymbol{u}(t)' + D\boldsymbol{u}(t) = N(\boldsymbol{u},t). \]
where the coefficient matrix are
\[ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix},\quad D = \begin{bmatrix} 0 & -1 & 0 \\ 0 & 0 & -1 \\ 0 & 0 & 0 \end{bmatrix},\]
and the nonlinear term is
\[ N(\boldsymbol{u},t) = -\frac{1}{2} \begin{bmatrix} 0 \\ 0 \\ f f'' \end{bmatrix}, \quad DN(\boldsymbol{u},t) = -\frac{1}{2} \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ f'' & 0 & f \end{bmatrix}.\]