Homework 4, due 08.00, 1/10-2015

Curvilinear coordinates

The files relevant to this homework reside in codes/fortran/HWK4.

In this homework you will compute derivatives and integrals on a logically rectangular domain (i.e. it has four sides and no holes) \(\Omega\). The strategy is to use a smooth mapping \((x,y)=(x(r,s),y(r,s))\) from the reference element \(\Omega_R = \{(r,s) \in [-1,1]^2\) } to \(\Omega\) to “transfer” the computations to the reference element.

For example if we want to approximate \(\frac{\partial u(x,y)}{\partial x}\) we use the chain rule to find

\[\begin{equation*} \frac{\partial u}{\partial x} = \frac{\partial r}{\partial x}\frac{\partial u}{\partial r}+\frac{\partial s}{\partial x}\frac{\partial u}{\partial s}, \end{equation*}\]

and similarly for

\[\begin{equation*} \frac{\partial u}{\partial y} = \frac{\partial r}{\partial y}\frac{\partial u}{\partial r}+\frac{\partial s}{\partial y}\frac{\partial u}{\partial s}. \end{equation*}\]

We can then introduce a Cartesian grid on the reference element:

hr = 2.d0/dble(nr)
hs = 2.d0/dble(ns)
do i = 0,nr
   r(i) = -1.d0 + dble(i)*hr
end do
do i = 0,ns
   s(i) = -1.d0 + dble(i)*hs
end do

to find \(u_r\) and \(u_s\) by standard finite difference formulas (see the subroutine differentiate.) To find the metric \(r_x, r_y, s_x, s_y\) we can first compute \(x_r, x_s, y_r, y_s\) and then use the above formulas with \(u = x\) and \(u = y\). This yields

\[\begin{equation*} \left[ \begin{array}{cc} r_x & s_x \\ r_y & s_y \end{array} \right] \left[ \begin{array}{cc} x_r & y_r \\ x_s & y_s \end{array} \right] = \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right]. \end{equation*}\]

It is also possible to compute integrals on the reference element. For example, we have that

\[\begin{equation*} \int_{\Omega} f(x,y) dxdy = \int_{-1}^1 \int_{-1}^1 f(x(r,s),y(r,s)) \, J(r,s) \, d r d s, \end{equation*}\]

where \(J(r,s) = x_r y_s - x_s y_r\) is the surface element. The second integrand can be approximated on the reference element, for example with the trapezoidal rule.

Assignments:

  1. Make and run the program homework4.f90 and use the Matlab script (or some other plotting tool) to display the grid.

  2. Use the above formula to find (compute numerically) the metric \(r_x, r_y, s_x, s_y\).

  3. Check that the results from 1) are correct by changing the mapping in xycoord.f90 to map the reference element into some geometrical shape for which you know the area (for example a sector of an annulus) and compute it on the reference element using the trapezoidal rule.

  4. Compute \(u_x\) and \(u_y\) for some different functions \(u(x,y)\) and some different mappings. Write a subroutine that approximates the error

    \[\begin{equation*} e_2(h_r,h_s) = \left(\int_{\Omega} \left(u_x(x,y)+u_y(x,y) - \left[(u_{\rm exact})_x + (u_{\rm exact})_y \right] \right)^2 dxdy \right)^{1/2}, \end{equation*}\]

    and plot the error as a function of the grid spacing for the different functions and the different mappings.

  5. Use the chain rule to find an expression for \(\Delta u = u_{xx}+u_{yy}\). Discretize it and repeat the experiments above (optional).

  6. Write up your findings neatly as a report and check it in to your repository.

Sample results

Below we give some sample results, note that the errors are messured in the max-norm so the results you produce are not going to be identical. Also note that the errors are plotted as a function of an effective gridsize \(h_{\rm eff} = \sqrt{h_r h_s \max J}\) Here we use 3 combinations of grids and functions:

The three combinations are

! Combination 1
x_coord = r+0.1d0*s
y_coord = s
u = sin(xc)*cos(yc)

! Combination 2
x_coord = (2.d0+r+0.1d0*sin(5.d0*pi*s))*cos(0.5d0*pi*s)
y_coord = (2.d0+r+0.1d0*sin(5.d0*pi*s))*sin(0.5d0*pi*s)
u = exp(xc+yc)

! Combination 3
x_coord = r
y_coord = s + s*r**2
u = xc**2+yc**2

And the grids and results are:

_images/hwk4grid1.png _images/hwk4combo1.png _images/hwk4grid2.png _images/hwk4combo2.png _images/hwk4grid3.png _images/hwk4combo3.png