Summary* Here we will present two methods for upwinding for the simplest convection problem. We will learn about Characteristics-Galerkin and Discontinuous-Galerkin Finite Element Methods.*

Let \(\Omega\) be the unit disk centered at 0; consider the rotation vector field $$ \bm{u} = [u1,u2], \qquad u_1 = y,\quad u_2 = -x.$$ Pure convection by \(u\) is $$ \partial_t c + \bm{u}.\nabla c = 0 \hbox{ in } \Omega\times(0,T) c (t=0) = c ^0 \hbox{ in } \Omega. $$ The exact solution \(c(x_t,t)\) at time \(t\) en point \(x_t\) is given by $$ c(x_t,t)=c^0(x,0) $$ where \(x_t\) is the particle path in the flow starting at point \(x\) at time \(0\). So \(x_t\) are solutions of $$ \dot{x_t} = u(x_t), \vec , \quad\ x_{t=0} =x , \quad\mbox{where}\quad \dot{x_t} = \frac{~d ( t \mapsto x_t)}{~d t} $$ The ODE are reversible and we want the solution at point \(x\) at time \(t\) ( not at point \(x_t\)) the initial point is \(x_{-t}\), and we have $$ c(x,t)=c^0(x_{-t},0) $$ The game consists in solving the equation until \(T=2\pi\), that is for a full revolution and to compare the final solution with the initial one; they should be equal.

Solution by a Characteristics-Galerkin Method In `FreeFem++`

there is an operator called \tt convect([u1,u2],dt,c) which compute
\( c\circ X\) with \(X\) is the convect field defined by
\( X(x)= x_{dt}\) and where \(x_\tau\) is particule path in the steady state velocity field \(\bm{u}=[u1,u2]\)
starting at point \(x\) at time \(\tau=0\), so \(x_\tau\) is solution of the following ODE:
$$ \dot{x}_\tau = u(x_\tau), \vec x_{\tau=0}=x. $$

When \(\bm{u}\) is piecewise constant; this is possible because \(x_\tau\) is then a polygonal curve which can be computed exactly and the solution exists always when \(u\) is divergence free; convect returns \(c(x_{df})=C\circ X\).

Example[convects.edp] // file convects.edp border C(t=0, 2*pi) { x=cos(t); y=sin(t); }; mesh Th = buildmesh(C(100)); fespace Uh(Th,P1); Uh cold, c = exp(-10*((x-0.3)^2 +(y-0.3)^2)); real dt = 0.17,t=0; Uh u1 = y, u2 = -x; for (int m=0; m<2*pi/dt ; m++) { t += dt; cold=c; c=convect([u1,u2],-dt,cold); plot(c,cmm=" t="+t + ", min=" + c[].min + ", max=" + c[].max); }

Remark: 3D plots can be done by adding the qualifyer "dim=3" to the plot instruction.

The method is very powerful but has two limitations: a/ it is not conservative, b/ it may diverge in rare cases when \(|u|\) is too small due to quadrature error.

Solution by Discontinuous-Galerkin FEM

Discontinuous Galerkin methods take advantage of the discontinuities of \(c\) at the edges to build upwinding. There are may formulations possible. We shall implement here the so-called dual-\(P_1^{DC}\) formulation (see Ern[ern]): $$ \int_\Omega(\frac{c^{n+1}-c^n}{\delta t} +u\cdot\nabla c)w +\int_E(\alpha|n\cdot u|-\frac 12 n\cdot u)[c]w =\int_{E_\Gamma^-}|n\cdot u| cw \forall w $$ where \(E\) is the set of inner edges and \(E_\Gamma^-\) is the set of boundary edges where \(u\cdot n<0\) (in our case there is no such edges). Finally \([c]\) is the jump of \(c\) across an edge with the convention that \(c^+\) refers to the value on the right of the oriented edge. Example[convects_end.edp]

// file convects.edp ... fespace Vh(Th,P1dc); Vh w, ccold, v1 = y, v2 = -x, cc = exp(-10*((x-0.3)^2 +(y-0.3)^2)); real u, al=0.5; dt = 0.05; macro n() (N.x*v1+N.y*v2) // Macro without parameter problem Adual(cc,w) = int2d(Th)((cc/dt+(v1*dx(cc)+v2*dy(cc)))*w) + intalledges(Th)((1-nTonEdge)*w*(al*abs(n)-n/2)*jump(cc)) // - int1d(Th,C)((n<0)*abs(n)*cc*w) // unused because cc=0 on $\partial\Omega^-$ - int2d(Th)(ccold*w/dt); for ( t=0; t< 2*pi ; t+=dt) { ccold=cc; Adual; plot(cc,fill=1,cmm="t="+t + ", min=" + cc[].min + ", max=" + cc[].max); }; real [int] viso=[-0.2,-0.1,0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1,1.1]; plot(c,wait=1,fill=1,ps="convectCG.eps",viso=viso); plot(c,wait=1,fill=1,ps="convectDG.eps",viso=viso);

Notice the new keywords, `intalledges`

to integrate on all edges of all triangles
\begin{equation}
\mathtt{intalledges}(\mathtt{Th}) \equiv \sum_T\in\mathtt{Th}\int_\partial T
\end{equation}

(so all internal edges are see two times ), nTonEdge which is one if the triangle has a boundary edge and zero otherwise, \tt jump to implement \([c]\). Results of both methods are shown on Figure [figconvect] with identical levels for the level line; this is done with the plot-modifier viso. #plotviso=

Notice also the macro where the parameter \(u\) is not used (but the syntax needs one) and which ends with a //; it simply replaces the name \tt n by \tt (N.x*v1+N.y*v2). As easily guessed \tt N.x,N.y is the normal to the edge.

[htbp]

[figconvect] The rotated hill after one revolution, left with Characteristics-Galerkin, on the right with Discontinuous \(P_1\) Galerkin FEM.

Now if you think that DG is too slow try this

// the same DG very much faster varf aadual(cc,w) = int2d(Th)((cc/dt+(v1*dx(cc)+v2*dy(cc)))*w) + intalledges(Th)((1-nTonEdge)*w*(al*abs(n)-n/2)*jump(cc)); varf bbdual(ccold,w) = - int2d(Th)(ccold*w/dt); matrix AA= aadual(Vh,Vh); matrix BB = bbdual(Vh,Vh); set (AA,init=t,solver=sparsesolver); Vh rhs=0; for ( t=0; t< 2*pi ; t+=dt) { ccold=cc; rhs[] = BB* ccold[]; cc[] = AA^-1*rhs[]; plot(cc,fill=0,cmm="t="+t + ", min=" + cc[].min + ", max=" + cc[].max); };

Notice the new keyword set to specify a solver in this framework; the modifier init is used to tel the solver that the matrix has not changed (init=true), and the name parameter are the same that in problem definition (see. [defproblem]) .

Finite Volume Methods can also be handled with `FreeFem++`

but it requires programming.
For instance the \(P_0-P_1\) Finite Volume Method of Dervieux et al associates to each \(P_0\)
function \(c^1\) a \(P_0\) function \(c^0\) with constant value around each vertex \(q^i\) equal to \(c^1(q^i)\)
on the cell \(\sigma_i\) made by all the medians of all triangles having \(q^i\) as vertex.
Then upwinding is done by taking left or right values at the median:
$$ \int_{\sigma_i}\frac 1{\delta t}({c^1}^{n+1}-{c^1}^n) + \int_{\partial\sigma_i}u\cdot n c^-=0 \forall i $$
It can be programmed as

load "mat_dervieux"; // external module in C++ must be loaded border a(t=0, 2*pi){ x = cos(t); y = sin(t); } mesh th = buildmesh(a(100)); fespace Vh(th,P1); Vh vh,vold,u1 = y, u2 = -x; Vh v = exp(-10*((x-0.3)^2 +(y-0.3)^2)), vWall=0, rhs =0; real dt = 0.025; // qf1pTlump means mass lumping is used problem FVM(v,vh) = int2d(th,qft=qf1pTlump)(v*vh/dt) - int2d(th,qft=qf1pTlump)(vold*vh/dt) + int1d(th,a)(((u1*N.x+u2*N.y)<0)*(u1*N.x+u2*N.y)*vWall*vh) + rhs[] ; matrix A; MatUpWind0(A,th,vold,[u1,u2]); for ( int t=0; t< 2*pi ; t+=dt){ vold=v; rhs[] = A * vold[] ; FVM; plot(v,wait=0); };

the mass lumping parameter forces a quadrature formula with Gauss points at the vertices so as to make the mass matrix diagonal; the linear system solved by a conjugate gradient method for instance will then converge in one or two iterations.

The right hand side \tt rhs is computed by an external C++ function#externalC++function \tt MatUpWind0(...) which is programmed as

// computes matrix a on a triangle for the Dervieux FVM int fvmP1P0(double q[3][2], // the 3 vertices of a triangle T double u[2], // convection velocity on T double c[3], // the P1 function on T double a[3][3],// output matrix double where[3] ) // where>0 means we're on the boundary { for(int i=0;i<3;i++) for(int j=0;j<3;j++) a[i][j]=0; for(int i=0;i<3;i++){ int ip = (i+1)%3, ipp =(ip+1)%3; double unL =-((q[ip][1]+q[i][1]-2*q[ipp][1])*u[0] -(q[ip][0]+q[i][0]-2*q[ipp][0])*u[1])/6; if(unL>0) { a[i][i] += unL; a[ip][i]-=unL;} else{ a[i][ip] += unL; a[ip][ip]-=unL;} if(where[i]\)||

unL=((q[ip][1]-q[i][1])*u[0] -(q[ip][0]-q[i][0])*u[1])/2; if(unL>0) { a[i][i]+=unL; a[ip][ip]+=unL;} } } return 1; }

It must be inserted into a larger .cpp file, shown in Appendix A,

` which is the load module linked to ``FreeFem++`

.