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3.4 Implementation and solution


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Substituting the overlap integrals (3.3#eq.4) into (3.2#eq.5) yields the FEM scheme

$\displaystyle \left(\begin{array}{c}
(1-p)\frac{\Delta x}{4}
-\theta\Delta t\fr...
...theta\Delta t\frac{u}{2}
-\theta\Delta t\frac{D}{\Delta x}
\end{array}\right)^T$ $\displaystyle \cdot$ $\displaystyle \left(\begin{array}{c}
f_{i-1}^{t+\Delta t} \\ f_{i }^{t+\Delta t} \\ f_{i+1}^{t+\Delta t}
\end{array}\right)
=$  
$\displaystyle \left(\begin{array}{c}
(1-p)\frac{\Delta x}{4}
-(\theta-1)\Delta ...
...)\Delta t\frac{u}{2}
-(\theta-1)\Delta t\frac{D}{\Delta x}
\end{array}\right)^T$ $\displaystyle \cdot$ $\displaystyle \left(\begin{array}{c}
f_{i-1}^{t} \\ f_{i }^{t} \\ f_{i+1}^{t}
\end{array}\right)$ (1)

Following an initialization where the initial condition is discretized with trivial projection on piecewise linear roof-top functions, the advection-diffusion scheme is implemented as
      BandMatrix a = new BandMatrix(3, f.length);
      BandMatrix b = new BandMatrix(3, f.length);
      double[]   c = new double[f.length];
      double h   = dx[0];
      double htm = h*(1-tune)/4;
      double htp = h*(1+tune)/4;

      for (int j=0; j<=n; j++) {
        a.setL(j,   htm +h*(-0.5*beta -alpha)* theta    );
        a.setD(j,2*(htp +h*(           alpha)* theta   ));
        a.setR(j,   htm +h*( 0.5*beta -alpha)* theta    );
        b.setL(j,   htm +h*(-0.5*beta -alpha)*(theta-1) );
        b.setD(j,2*(htp +h*(           alpha)*(theta-1)));
        b.setR(j,   htm +h*( 0.5*beta -alpha)*(theta-1) );
      }
      c=b.dot(f);                                 //Right hand side
      c[0]=c[0]+b.getL(0)*f[n];                   // with periodicity
      c[n]=c[n]+b.getR(n)*f[0];

      fp=a.solve3(c);                             //Solve linear problem
Here again, the BandMatrix.solve3() method is used to compute a direct solution in $ \mathcal{O}(N)$ operations, using an LU-factorization that will be examined with more details in the coming section. The entire scheme is illustrated below with the advection of a box function, calculated using a piecewise linear FEM discretization (with $ p=1/3$ ) and a partly implicit time integration (with $ \theta=0.55$ ).
JBONE applet:  press Start/Stop to simulate the advection of a box function, so as to test how the linear finite elements scheme affects the dispersion and the damping of short and long wavelengths superposed in a box function.



Numerical experiments: tunable finite-elements
  1. Vary the TimeImplicit parameter $ \theta\in[\frac{1}{2};1]$ and the TuneIntegr parameter $ p\in]0;1]$ to determine which combination has the smallest numerical damping.
  2. Repeat the study and try to minimize the phase errors.
  3. Introduce a finite physical Diffusion=1 and adjust the TimeImplicit and the TuneIntegr parameters to calculate a solution that is physically meaningful even with a large CFL number.

Having spent a considerable effort understanding the FEM formulation, isn't it frustrating to see how similar the code is to the implicit Crank-Nicholson FD scheme in sect.2.5? This should be the main argument for the community who still uses implicit finite difference schemes: with a little bit of thinking but the same computational cost, the finite element scheme is now considerably more flexibile: it is for example easy to densify the mesh16, modify the partly implicit time integration from centered to fully implicit $ \theta\in[\frac{1}{2};1]$ and tune the integration $ p\in[0;1]$ . Convince yourself that Crank-Nicholson (2.5.1#eq.12) and this FEM scheme (3.4#eq.1) are indeed equivalent when using a homogeneous mesh $ x_j=j\Delta x$ , a time centered integration $ \theta=\frac{1}{2}$ and a trapezoidal quadrature $ p=1$ .

SYLLABUS  Previous: 3.3 Numerical quadrature  Up: 3 FINITE ELEMENT METHOD  Next: 3.5 Linear solvers

      
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