CppNoddy: include/PDE_double_IBVP.h File Reference

A specification of a class for an $n^{th}$ -order IBVP of the form. More...

#include <DenseVector.h>
#include <DenseMatrix.h>
#include <BandedMatrix.h>
#include <Equation_3matrix.h>
#include <Residual_with_coords.h>
#include <TwoD_Node_Mesh.h>
#include <Uncopyable.h>
#include <Timer.h>

Go to the source code of this file.

Classes
class	CppNoddy::PDE_double_IBVP< _Type >
	A templated object for real/complex vector system of unsteady equations. More...

Namespaces
namespace	CppNoddy
	A collection of OO numerical routines aimed at simple (typical) applied problems in continuum mechanics.

Detailed Description

A specification of a class for an $n^{th}$ -order IBVP of the form.

$M_2( {\underline f}(y,x,t), y, x, t )\cdot {\underline f}_x (y,x,t) + M_1( {\underline f}(y,x,t), y,x, t )\cdot {\underline f}_t (y,x,t) + M_0( {\underline f}(y,x,t), y,x, t )\cdot {\underline f}_y (y,x,t) = {\underline R}( {\underline f}(y,x,t), y,x, t )\,,$

subject to conditions defined at $y = y_{bottom}$ and $y_{top}$ for some components of ${\underline f}(y,x,t)$ . Here is a matrix for the x-variation and is the mass matrix for the t-variation, whilst is not restricted to be an identity matrix (though it often will be). The solution at the new time step $t+\Delta t$ and spatial location is

${\underline f}^{new} = {\underline F}_{j} + {\underline g}$

where ${\underline F}_{j}$ is the current guess at the solution at this point and ${\underline g}$ is the linearised correction. The solution at the previous time at $x = x_{j}$ is

${\underline f}^{old} = {\underline O}_{j}$

A Crank-Nicolson method is employed with the linearised problem at the mid-t-point $t + \Delta t /2$ and mid-x-point $( x_{j+1} + x_{j} ) / 2$ being:

$\frac{2}{x_{j+1} - x_{j}}\, M_2 \cdot {\underline g } + \frac{2}{\Delta t}\, M_1 \cdot {\underline g } + M_0 \cdot {\underline g}_y - J \cdot {\underline g}$

$+ J_{2} \cdot \frac{ ( \underline F_{j+1} + \underline O_{j+1} - \underline F_j - \underline O_j) }{ 2(x_{j+1} - x_j) } \cdot {\underline g}$

$+ J_{1} \cdot \frac{(\underline F_{j+1} + \underline F_j - \underline O_j - \underline O_{j+1})}{2\Delta t} \cdot {\underline g}$

$+ J_{0} \cdot \frac{( \underline F_{j+1} + \underline F_j + \underline O_j + \underline O_{j+1})_y}{4} \cdot {\underline g}$

$= 4 {\underline R} - M_0 \cdot \biggl ({\underline F_j}_y + {\underline O_j}_y + {\underline F_{j+1}}_y + {\underline O_{j+1}}_y \biggr ) - \frac{2}{x_{j+1}-x_j} M_1 \cdot \biggl ( {\underline F}_{j+1} + {\underline O}_{j+1} - {\underline F}_{j} - {\underline O}_{j} \biggr ) - \frac{2}{\Delta t} M_2 \cdot \biggl ( {\underline F}_{j+1} + {\underline F}_{j} - {\underline O}_{j+1} - {\underline O}_{j} \biggr )$

Where $M_{0,1,2}, J, J_{0,1,2}, R$ are evaluated at the mid-t/mid-x step with arguments $\left ( \frac{\underline F_{j} + \underline O_{j} + \underline F_{j+1} + \underline O_{j+1} }{4}, y, ( x_{j+1} + x_{j} ) / 2, t + \frac{\Delta t}{2} \right )$ , with $J_{0,1,2}$ denoting the Jacobian of the mass matrices $\partial {M_{0,1,2}}_{ij} / \partial f_k$ . This problem is solved by second-order central differencing the equation at the spatial ( ) inter-node mid points.

Definition in file PDE_double_IBVP.h.

Classes

Namespaces

Detailed Description