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Pure Convection : The Rotating Hill

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 $$(0,0)$$; consider the rotation vector field

$\mathbf{u} = [u1,u2], \qquad u_1 = y,\quad u_2 = -x$

Pure convection by $$\mathbf{u}$$ is

$\begin{split}\begin{array}{rcl} \partial_t c + \mathbf{u}.\nabla c &= 0 &\hbox{ in } \Omega\times(0,T)\\ c (t=0) &= c ^0 &\hbox{ in } \Omega. \end{array}\end{split}$

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

$\dot{x_t} = u(x_t), \quad\ x_{t=0} =x , \quad\mbox{where}\quad \dot{x_t} = \frac{\text{d} ( t \mapsto x_t)}{\text{d} t}$

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 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 $$\mathbf{u}=[u1,u2]$$ starting at point $$x$$ at time $$\tau=0$$, so $$x_\tau$$ is solution of the following ODE:

$\dot{x}_\tau = u(x_\tau), \mathbf{x}_{\tau=0}=x.$

When $$\mathbf{u}$$ 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 $$\mathbf{u}$$ is divergence free; convect returns $$c(x_{df})=C\circ X$$.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 // Parameters real dt = 0.17; // Mesh border C(t=0., 2.*pi) {x=cos(t); y=sin(t);}; mesh Th = buildmesh(C(100)); // Fespace fespace Uh(Th, P1); Uh cold, c = exp(-10*((x-0.3)^2 +(y-0.3)^2)); Uh u1 = y, u2 = -x; // Time loop real t = 0; 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); } 

Note

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

The method is very powerful but has two limitations:

• it is not conservative
• it may diverge in rare cases when $$|\mathbf{u}|$$ 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 [ERN2006]):

$\int_\Omega(\frac{c^{n+1}-c^n}{\delta t} +u\cdot\nabla c)w +\int_E(\alpha|n\cdot u|-\frac 12 n\cdot u)[c]w =\int_{E_\Gamma^-}|n\cdot u| cw~~~\forall w$

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.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 // Parameters real al=0.5; real dt = 0.05; // Mesh border C(t=0., 2.*pi) {x=cos(t); y=sin(t);}; mesh Th = buildmesh(C(100)); // Fespace fespace Vh(Th,P1dc); Vh w, ccold, v1 = y, v2 = -x, cc = exp(-10*((x-0.3)^2 +(y-0.3)^2)); // Macro macro n() (N.x*v1 + N.y*v2) // Macro without parameter // Problem 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) ) - int2d(Th)( ccold*w/dt ) ; // Time iterations for (real t = 0.; t < 2.*pi; t += dt){ ccold = cc; Adual; plot(cc, fill=1, cmm="t="+t+", min="+cc[].min+", max="+ cc[].max); } // Plot 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(cc, wait=1, fill=1, ps="ConvectCG.eps", viso=viso); plot(cc, wait=1, fill=1, ps="ConvectDG.eps", viso=viso); 

Note

New keywords: intalledges to integrate on all edges of all triangles

$\mathtt{intalledges}(\mathtt{Th}) \equiv \sum_{T\in\mathtt{Th}}\int_{\partial T }$

(so all internal edges are see two times), nTonEdge which is one if the triangle has a boundary edge and two otherwise, jump to implement $$[c]$$.

Results of both methods are shown on Fig. 17 nad Fig. 18 with identical levels for the level line; this is done with the plot-modifier viso.

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

Fig. 17 The rotating hill after one revolution with Characteristics-Galerkin

Fig. 18 The rotating hill after one revolution with Discontinuous $$P_1$$ Galerkin

Rotating hill

Now if you think that DG is too slow try this:

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 // Parameters real al=0.5; real dt = 0.05; // Mesh border C(t=0., 2.*pi) {x=cos(t); y=sin(t);}; mesh Th = buildmesh(C(100)); // Fespace fespace Vh(Th,P1dc); Vh w, ccold, v1 = y, v2 = -x, cc = exp(-10*((x-0.3)^2 +(y-0.3)^2)); Vh rhs=0; // Macro macro n() (N.x*v1 + N.y*v2) // Macro without parameter // Problem real t = 0.; varf vAdual (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 vBdual (cc, w) = - int2d(Th)( ccold*w/dt ) ; matrix AA = vAdual(Vh, Vh); matrix BB = vBdual(Vh, Vh); set (AA, init=t, solver=sparsesolver); // Time iterations for (t = 0.; t < 2.*pi; t += dt){ ccold = cc; rhs[] = BB * ccold[]; cc[] = AA^-1 * rhs[]; plot(cc, fill=1, 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 tell the solver that the matrix has not changed (init=true), and the name parameter are the same that in problem definition (see Problem)

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 :

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 load "mat_dervieux"; //External module in C++ must be loaded // Parameters real dt = 0.025; // Mesh border a(t=0., 2.*pi){x=cos(t); y=sin(t);} mesh th = buildmesh(a(100)); // Fespace 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; // Problem //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]); // Time loop 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 rhs is computed by an external C++ function MatUpWind0(...) which is programmed as :

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 // 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] && where[ip]){ // this is a boundary edge 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.