Пример #1
0
// GAUSS-LOBATTO QUADRATURE ALONG EDGE
void SetEdgeDataGL_Unst(const mesh& Mesh, 
			int NumQuadPoints, 
			int NumBasisOrder, 
			edge_data_Unst* EdgeData)
{
  // Quick error check
  if (NumQuadPoints<2 || NumQuadPoints>6 || NumBasisOrder<1 || NumBasisOrder>5)
    {
      printf(" \n");
      printf(" Error in SetEdgeData_Unst.cpp \n");
      printf("   NumQuadPoints must be 2,3,4,5, or 6.\n");
      printf("   NumBasisOrder must be 1,2,3,4, or 5.\n");
      printf("     NumQuadPoints = %i\n",NumQuadPoints);
      printf("     NumBasisOrder = %i\n",NumBasisOrder);
      printf("\n");
      exit(1);
    }

  // ---------------------------------
  // Set quadrature weights and points
  // ---------------------------------
  switch( NumQuadPoints )
    {
    case 2:
      EdgeData->GL_wgts1d->set(1,  1.0 );
      EdgeData->GL_wgts1d->set(2,  1.0 );
      
      EdgeData->GL_xpts1d->set(1,  1.0 );
      EdgeData->GL_xpts1d->set(2, -1.0 );
      break;
      
    case 3:
      EdgeData->GL_wgts1d->set(1,  onethird );
      EdgeData->GL_wgts1d->set(2,  4.0*onethird );
      EdgeData->GL_wgts1d->set(3,  onethird );
      
      EdgeData->GL_xpts1d->set(1,  1.0 );
      EdgeData->GL_xpts1d->set(2,  0.0 );
      EdgeData->GL_xpts1d->set(3, -1.0 );
      break;
      
    case 4:
      EdgeData->GL_wgts1d->set(1,  0.5*onethird );
      EdgeData->GL_wgts1d->set(2,  2.5*onethird );
      EdgeData->GL_wgts1d->set(3,  2.5*onethird );
      EdgeData->GL_wgts1d->set(4,  0.5*onethird );
      
      EdgeData->GL_xpts1d->set(1,  1.0  );
      EdgeData->GL_xpts1d->set(2,  osq5 );
      EdgeData->GL_xpts1d->set(3, -osq5 );
      EdgeData->GL_xpts1d->set(4, -1.0  );
      break;
      
    case 5:
      EdgeData->GL_wgts1d->set(1,  0.1  );
      EdgeData->GL_wgts1d->set(2,  49.0/90.0 );
      EdgeData->GL_wgts1d->set(3,  32.0/45.0 );
      EdgeData->GL_wgts1d->set(4,  49.0/90.0 );
      EdgeData->GL_wgts1d->set(5,  0.1 );
      
      EdgeData->GL_xpts1d->set(1,  1.0      );
      EdgeData->GL_xpts1d->set(2,  sq3*osq7 );
      EdgeData->GL_xpts1d->set(3,  0.0      );
      EdgeData->GL_xpts1d->set(4, -sq3*osq7 );
      EdgeData->GL_xpts1d->set(5, -1.0      );        
      break;
      
    case 6:      
      EdgeData->GL_wgts1d->set(1,  0.2*onethird  );
      EdgeData->GL_wgts1d->set(2,  (1.4 - 0.1*sq7)*onethird );
      EdgeData->GL_wgts1d->set(3,  (1.4 + 0.1*sq7)*onethird );
      EdgeData->GL_wgts1d->set(4,  (1.4 + 0.1*sq7)*onethird );
      EdgeData->GL_wgts1d->set(5,  (1.4 - 0.1*sq7)*onethird );
      EdgeData->GL_wgts1d->set(6,  0.2*onethird );
      
      EdgeData->GL_xpts1d->set(1,  1.0                           );
      EdgeData->GL_xpts1d->set(2,  (1/21.0)*sqrt(147.0+42.0*sq7) );
      EdgeData->GL_xpts1d->set(3,  (1/21.0)*sqrt(147.0-42.0*sq7) );
      EdgeData->GL_xpts1d->set(4, -(1/21.0)*sqrt(147.0-42.0*sq7) );
      EdgeData->GL_xpts1d->set(5, -(1/21.0)*sqrt(147.0+42.0*sq7) );
      EdgeData->GL_xpts1d->set(6, -1.0                           );
      break;
    }

  // ---------------------------------
  // Legendre basis functions on the 
  // left and right of each edge
  // ---------------------------------
  const int NumEdges = Mesh.get_NumEdges();
  const int NumBasisComps = (NumBasisOrder*(NumBasisOrder+1))/2;
  dTensor1 xp1(3);
  dTensor1 yp1(3);
  dTensor1 xp2(3);
  dTensor1 yp2(3);
  dTensor1 xy1(2);
  dTensor1 xy2(2);
  dTensor1 mu1(NumBasisComps);
  dTensor1 mu2(NumBasisComps);

  for (int i=1; i<=NumEdges; i++)
    {   
      // Get edge information
      const double x1 = Mesh.get_edge(i,1);
      const double y1 = Mesh.get_edge(i,2);
      const double x2 = Mesh.get_edge(i,3);
      const double y2 = Mesh.get_edge(i,4);
      
      const int e1 = Mesh.get_eelem(i,1);
      const int e2 = Mesh.get_eelem(i,2);

      // Get element information about
      // the two elements that meet at
      // the current edge
      const double Area1 = Mesh.get_area_prim(e1);
      const double Area2 = Mesh.get_area_prim(e2);

      for (int k=1; k<=3; k++)
	{
	  xp1.set(k, Mesh.get_node(Mesh.get_tnode(e1,k),1) );
	  yp1.set(k, Mesh.get_node(Mesh.get_tnode(e1,k),2) );

	  xp2.set(k, Mesh.get_node(Mesh.get_tnode(e2,k),1) );
	  yp2.set(k, Mesh.get_node(Mesh.get_tnode(e2,k),2) );
	}

      const double xc1 = (xp1.get(1) + xp1.get(2) + xp1.get(3))/3.0;
      const double yc1 = (yp1.get(1) + yp1.get(2) + yp1.get(3))/3.0;
      const double xc2 = (xp2.get(1) + xp2.get(2) + xp2.get(3))/3.0;
      const double yc2 = (yp2.get(1) + yp2.get(2) + yp2.get(3))/3.0;

      // quadrature points on the edge
      for (int m=1; m<=NumQuadPoints; m++)
	{
	  // Take integration point s (in [-1,1])
	  // and map to physical domain
	  const double s = EdgeData->GL_xpts1d->get(m);
	  const double x = x1 + 0.5*(s+1.0)*(x2-x1);
	  const double y = y1 + 0.5*(s+1.0)*(y2-y1);

	  // Take physical point (x,y)
	  // and map into the coordinates
	  // of the two triangles that are
	  // adjacent to the current edge
	  xy1.set(1, ((yp1.get(3)-yp1.get(1))*(x-xc1) 
		    + (xp1.get(1)-xp1.get(3))*(y-yc1))/(2.0*Area1) );
	  xy1.set(2, ((yp1.get(1)-yp1.get(2))*(x-xc1) 
		    + (xp1.get(2)-xp1.get(1))*(y-yc1))/(2.0*Area1) );
	  
	  xy2.set(1, ((yp2.get(3)-yp2.get(1))*(x-xc2) 
		    + (xp2.get(1)-xp2.get(3))*(y-yc2))/(2.0*Area2) );
	  xy2.set(2, ((yp2.get(1)-yp2.get(2))*(x-xc2) 
		    + (xp2.get(2)-xp2.get(1))*(y-yc2))/(2.0*Area2) );

	  // Evaluate monomials at locations xy1
	  double xi = xy1.get(1);
	  double xi2 = xi*xi;
	  double xi3 = xi*xi2;
	  double xi4 = xi*xi3;

	  double eta = xy1.get(2);
	  double eta2 = eta*eta;
	  double eta3 = eta*eta2;
	  double eta4 = eta*eta3;

	  switch( NumBasisOrder )
	    {
	    case 5:  // fifth order		    		    
	      mu1.set(15, eta4     );
	      mu1.set(14, xi4      );
	      mu1.set(13, xi2*eta2 );
	      mu1.set(12, eta3*xi  );
	      mu1.set(11, xi3*eta  );
	      
	    case 4:  // fourth order
	      mu1.set(10, eta3     );
	      mu1.set(9,  xi3      );
	      mu1.set(8,  xi*eta2  );
	      mu1.set(7,  eta*xi2  );
	      
	    case 3:  // third order
	      mu1.set(6,  eta2     );
	      mu1.set(5,  xi2      );
	      mu1.set(4,  xi*eta   );		    
	      
	    case 2:  // second order		    
	      mu1.set(3, eta       );
	      mu1.set(2, xi        );
	      
	    case 1:  // first order
	      mu1.set(1, 1.0       );
	      
	      break;		    
	    }
	  
	  // Evaluate monomials at locations xy2
	  xi = xy2.get(1);
	  xi2 = xi*xi;
	  xi3 = xi*xi2;
	  xi4 = xi*xi3;
	  
	  eta = xy2.get(2);
	  eta2 = eta*eta;
	  eta3 = eta*eta2;
	  eta4 = eta*eta3;

	  switch( NumBasisOrder )
	    {
	    case 5:  // fifth order		    		    
	      mu2.set(15, eta4     );
	      mu2.set(14, xi4      );
	      mu2.set(13, xi2*eta2 );
	      mu2.set(12, eta3*xi  );
	      mu2.set(11, xi3*eta  );
	      
	    case 4:  // fourth order
	      mu2.set(10, eta3     );
	      mu2.set(9,  xi3      );
	      mu2.set(8,  xi*eta2  );
	      mu2.set(7,  eta*xi2  );
	      
	    case 3:  // third order
	      mu2.set(6,  eta2     );
	      mu2.set(5,  xi2      );
	      mu2.set(4,  xi*eta   );		    
	      
	    case 2:  // second order		    
	      mu2.set(3, eta       );
	      mu2.set(2, xi        );
	      
	    case 1:  // first order
	      mu2.set(1, 1.0       );
	      
	      break;		    
	    }
	  
	  // Finally, convert monomials to Legendre Polys
	  // on the two adjacent triangle
	  for (int k=1; k<=NumBasisComps; k++)
	    {
	      double tmp1 = 0.0;
	      double tmp2 = 0.0;
	      for (int j=1; j<=k; j++)
		{  
		  tmp1 = tmp1 + Mmat[k-1][j-1]*mu1.get(j);
		  tmp2 = tmp2 + Mmat[k-1][j-1]*mu2.get(j);
		}
	      
	      EdgeData->GL_phi_left->set(i,m,k,  tmp1 );
	      EdgeData->GL_phi_right->set(i,m,k, tmp2 );
	    }
	}
    }
  
}
Пример #2
0
// Right-hand side for hyperbolic PDE in divergence form
//
//       q_t = -( f(q,x,y,t)_x + g(q,x,y,t)_y ) + Psi(q,x,y,t)
//
void LaxWendroff_Unst(double dt,
    const mesh& Mesh, const edge_data_Unst& EdgeData,
    dTensor3& aux,                  // SetBndValues modifies ghost cells
    dTensor3& q,                    // SetBndValues modifies ghost cells
    dTensor3& Lstar, dTensor1& smax)
{

    const int NumElems      = Mesh.get_NumElems();
    const int NumPhysElems  = Mesh.get_NumPhysElems();
    const int NumEdges      = Mesh.get_NumEdges();
    const int meqn          = q.getsize(2);
    const int kmax          = q.getsize(3);
    const int maux          = aux.getsize(2);
    const int space_order   = dogParams.get_space_order();
    dTensor3 EdgeFluxIntegral(NumElems,meqn,kmax);
    dTensor3 ElemFluxIntegral(NumElems,meqn,kmax);
    dTensor3              Psi(NumElems,meqn,kmax);

    // ---------------------------------------------------------
    // Boundary Conditions
    SetBndValues_Unst(Mesh, &q, &aux);
    // ---------------------------------------------------------

    // --------------------------------------------------------------------- //
    // Part 0: Compute the Lax-Wendroff "flux" function:
    //
    // Here, we include the extra information about time derivatives.
    // --------------------------------------------------------------------- //
    dTensor3 F(NumElems, meqn, kmax );  F.setall(0.);
    dTensor3 G(NumElems, meqn, kmax );  G.setall(0.);
    L2ProjectLxW_Unst( dogParams.get_time_order(), 1.0, 0.5*dt, dt*dt/6.0, 1, NumElems,
        space_order, space_order, space_order, space_order, Mesh,
        &q, &aux, &F, &G, &FluxFunc, &DFluxFunc, &D2FluxFunc );

    // ---------------------------------------------------------
    // Part I: compute source term
    // --------------------------------------------------------- 
    if ( dogParams.get_source_term()>0 )
    {        
        // eprintf("error: have not implemented source term for LxW solver.");
        printf("Source term has not been implemented for LxW solver.  Terminating program.");
        exit(1);
    }
    Lstar.setall(0.);
    // ---------------------------------------------------------


    // ---------------------------------------------------------
    // Part II: compute flux integral on element edges
    // ---------------------------------------------------------

    // Loop over all interior edges
    EdgeFluxIntegral.setall(0.);
    ElemFluxIntegral.setall(0.);

#pragma omp parallel for
    // Loop over all interior edges
    for (int i=1; i<=NumEdges; i++)
    {
        // Edge coordinates
        double x1 = Mesh.get_edge(i,1);
        double y1 = Mesh.get_edge(i,2);
        double x2 = Mesh.get_edge(i,3);
        double y2 = Mesh.get_edge(i,4);

        // Elements on either side of edge
        int ileft  = Mesh.get_eelem(i,1);
        int iright = Mesh.get_eelem(i,2);  
        double Areal = Mesh.get_area_prim(ileft);
        double Arear = Mesh.get_area_prim(iright);

        // Scaled normal to edge
        dTensor1 nhat(2);      
        nhat.set(1, (y2-y1) );
        nhat.set(2, (x1-x2) );

        // Variables to store flux integrals along edge
        dTensor2 Fr_tmp(meqn,dogParams.get_space_order());
        dTensor2 Fl_tmp(meqn,dogParams.get_space_order());

        // Loop over number of quadrature points along each edge
        for (int ell=1; ell<=dogParams.get_space_order(); ell++)
        {
            dTensor1   Ql(meqn),   Qr(meqn);
            dTensor1  ffl(meqn),  ffr(meqn);  // << -- NEW PART -- >>
            dTensor1 Auxl(maux), Auxr(maux);

            // Riemann data - q
            for (int m=1; m<=meqn; m++)
            {
                Ql.set(m, 0.0 );
                Qr.set(m, 0.0 );

                // << -- NEW PART, ffl and ffr -- >> //
                ffl.set(m, 0.0 );
                ffr.set(m, 0.0 );

                for (int k=1; k<=kmax; k++)
                {
                    Ql.set(m, Ql.get(m) + EdgeData.phi_left->get(i,ell,k) 
                            *q.get(ileft, m,k) );
                    Qr.set(m, Qr.get(m) + EdgeData.phi_right->get(i,ell,k)
                            *q.get(iright,m,k) );

                    // << -- NEW PART, ffl and ffr -- >> //
                    // Is this the correct way to use the normal vector?
                    ffl.set(m, ffl.get(m) + EdgeData.phi_left->get (i, ell, k) * ( 
                        nhat.get(1)*F.get( ileft, m, k) + nhat.get(2)*G.get( ileft, m, k) ) );

                    ffr.set(m, ffr.get(m) + EdgeData.phi_right->get(i, ell, k) * (
                        nhat.get(1)*F.get(iright, m, k) + nhat.get(2)*G.get(iright, m, k) ) );

                }
            }

            // Riemann data - aux
            for (int m=1; m<=maux; m++)
            {
                Auxl.set(m, 0.0 );
                Auxr.set(m, 0.0 );

                for (int k=1; k<=kmax; k++)
                {
                    Auxl.set(m, Auxl.get(m) + EdgeData.phi_left->get(i,ell,k)  * aux.get(ileft, m,k) );
                    Auxr.set(m, Auxr.get(m) + EdgeData.phi_right->get(i,ell,k) * aux.get(iright,m,k) );
                }
            }

            // Solve Riemann problem
            dTensor1 xedge(2);
            double s = EdgeData.xpts1d->get(ell);
            xedge.set(1, x1 + 0.5*(s+1.0)*(x2-x1) );
            xedge.set(2, y1 + 0.5*(s+1.0)*(y2-y1) );

            // Solve the Riemann problem for this edge
            dTensor1 Fl(meqn), Fr(meqn);

            // Use the time-averaged fluxes to define left and right values for
            // the Riemann solver.
            const double smax_edge = RiemannSolveLxW(
                    nhat, xedge, Ql, Qr, Auxl, Auxr, ffl, ffr, Fl, Fr);
            smax.set(i, Max(smax_edge,smax.get(i)) );

            // Construct fluxes
            for (int m=1; m<=meqn; m++)
            {
                Fr_tmp.set(m,ell, Fr.get(m) );
                Fl_tmp.set(m,ell, Fl.get(m) );
            }
        }

        // Add edge integral to line integral around the full element
        for (int m=1; m<=meqn; m++)
        for (int k=1; k<=kmax; k++)
        {
            double Fl_sum = 0.0;
            double Fr_sum = 0.0;
            for (int ell=1; ell<=dogParams.get_space_order(); ell++)
            {
                Fl_sum = Fl_sum + 0.5*EdgeData.wgts1d->get(ell)
                    *EdgeData.phi_left->get(i,ell,k) *Fl_tmp.get(m,ell);
                Fr_sum = Fr_sum + 0.5*EdgeData.wgts1d->get(ell)
                    *EdgeData.phi_right->get(i,ell,k)*Fr_tmp.get(m,ell);
            }
            EdgeFluxIntegral.set(ileft, m,k, EdgeFluxIntegral.get(ileft, m,k) + Fl_sum/Areal );
            EdgeFluxIntegral.set(iright,m,k, EdgeFluxIntegral.get(iright,m,k) - Fr_sum/Arear );
        }
    }
    // ---------------------------------------------------------

    // ---------------------------------------------------------
    // Part III: compute intra-element contributions
    // ---------------------------------------------------------
    if( dogParams.get_space_order() > 1 )
    {
        L2ProjectGradAddLegendre_Unst(1, NumPhysElems, space_order, 
            Mesh, &F, &G, &ElemFluxIntegral );
    }
    // ---------------------------------------------------------

    // ---------------------------------------------------------
    // Part IV: construct Lstar
    // ---------------------------------------------------------
    if (dogParams.get_source_term()==0)  // Without Source Term
    { 
#pragma omp parallel for
        for (int i=1; i<=NumPhysElems; i++)	
        for (int m=1; m<=meqn; m++)
        for (int k=1; k<=kmax; k++)
        {
            double tmp = ElemFluxIntegral.get(i,m,k) - EdgeFluxIntegral.get(i,m,k);
            Lstar.set(i,m,k, tmp );	      
        }
    }
    else  // With Source Term
    {
#pragma omp parallel for
        for (int i=1; i<=NumPhysElems; i++)
        for (int m=1; m<=meqn; m++)
        for (int k=1; k<=kmax; k++)
        {
//          double tmp = ElemFluxIntegral.get(i,m,k) 
//              - EdgeFluxIntegral.get(i,m,k)
//              + Psi.get(i,m,k);

//          Lstar.set(i,m,k, tmp );

            printf("Source term has not been implemented for LxW solver.  Terminating program.");
            exit(1);
        }
    }
    // ---------------------------------------------------------

    // ---------------------------------------------------------
    // Part V: add extra contributions to Lstar
    // ---------------------------------------------------------
    // Call LstarExtra
    LstarExtra_Unst(Mesh, &q, &aux, &Lstar);
    // ---------------------------------------------------------

    // ---------------------------------------------------------
    // Part VI: artificial viscosity limiter
    // ---------------------------------------------------------  
//  if (dogParams.get_space_order()>1  &&
//          dogParams.using_viscosity_limiter())
//  {  ArtificialViscosity(&aux,&q,&Lstar);  }
    // ---------------------------------------------------------

}
Пример #3
0
void ConstructL_Unst(
    const double t,
    const dTensor2* vel_vec,
    const mesh& Mesh,
    const edge_data_Unst& EdgeData,
    dTensor3& aux, // SetBndValues_Unst modifies ghost cells
    dTensor3& q,   // SetBndValues_Unst modifies ghost cells
    dTensor3& Lstar, 
    dTensor1& smax)
{

    const int NumElems      = Mesh.get_NumElems();
    const int NumPhysElems  = Mesh.get_NumPhysElems();
    const int NumEdges      = Mesh.get_NumEdges();
    const int meqn          = q.getsize(2);
    const int kmax          = q.getsize(3);
    const int maux          = aux.getsize(2);
    const int space_order   = dogParams.get_space_order();

    dTensor3 EdgeFluxIntegral(NumElems,meqn,kmax);
    dTensor3 ElemFluxIntegral(NumElems,meqn,kmax);
    dTensor3              Psi(NumElems,meqn,kmax);


    // ---------------------------------------------------------
    // Boundary Conditions
    SetBndValues_Unst(Mesh,&q,&aux);  
    
    // Positivity limiter
    void ApplyPosLimiter_Unst(const mesh& Mesh, const dTensor3& aux, dTensor3& q);
    if( dogParams.using_moment_limiter() )
    { ApplyPosLimiter_Unst(Mesh, aux, q); }
    // ---------------------------------------------------------

    // ---------------------------------------------------------
    // Part I: compute flux integral on element edges
    // ---------------------------------------------------------

    // Loop over all interior edges and solve Riemann problems
    // dTensor1 nvec(2);

    // Loop over all interior edges
    EdgeFluxIntegral.setall(0.);
    ElemFluxIntegral.setall(0.);

    // Loop over all interior edges
#pragma omp parallel for
    for (int i=1; i<=NumEdges; i++)
    {
        // Edge coordinates
        double x1 = Mesh.get_edge(i,1);
        double y1 = Mesh.get_edge(i,2);
        double x2 = Mesh.get_edge(i,3);
        double y2 = Mesh.get_edge(i,4);

        // Elements on either side of edge
        int ileft  = Mesh.get_eelem(i,1);
        int iright = Mesh.get_eelem(i,2);  
        double Areal = Mesh.get_area_prim(ileft);
        double Arear = Mesh.get_area_prim(iright);

        // Scaled normal to edge
        dTensor1 nhat(2);      
        nhat.set(1, (y2-y1) );
        nhat.set(2, (x1-x2) );

        // Variables to store flux integrals along edge
        dTensor2 Fr_tmp(meqn,dogParams.get_space_order());
        dTensor2 Fl_tmp(meqn,dogParams.get_space_order());

        // Loop over number of quadrature points along each edge
        for (int ell=1; ell<=dogParams.get_space_order(); ell++)
        {
            dTensor1 Ql(meqn),Qr(meqn);
            dTensor1 Auxl(maux),Auxr(maux);	  

            // Riemann data - q
            for (int m=1; m<=meqn; m++)
            {
                Ql.set(m, 0.0 );
                Qr.set(m, 0.0 );

                for (int k=1; k<=kmax; k++)
                {
                    Ql.set(m, Ql.get(m) + EdgeData.phi_left->get(i,ell,k) 
                            *q.get(ileft, m,k) );
                    Qr.set(m, Qr.get(m) + EdgeData.phi_right->get(i,ell,k)
                            *q.get(iright,m,k) );
                }

            }


            // Riemann data - aux
            for (int m=1; m<=maux; m++)
            {
                Auxl.set(m, 0.0 );
                Auxr.set(m, 0.0 );

                for (int k=1; k<=kmax; k++)
                {
                    Auxl.set(m, Auxl.get(m) + EdgeData.phi_left->get(i,ell,k)
                            *aux.get(ileft, m,k) );
                    Auxr.set(m, Auxr.get(m) + EdgeData.phi_right->get(i,ell,k)
                            *aux.get(iright,m,k) );
                }
            }

            // Solve Riemann problem
            dTensor1 xedge(2);
            double s = EdgeData.xpts1d->get(ell);
            xedge.set(1, x1 + 0.5*(s+1.0)*(x2-x1) );
            xedge.set(2, y1 + 0.5*(s+1.0)*(y2-y1) );
            dTensor1 Fl(meqn),Fr(meqn);
            const double smax_edge = RiemannSolve(vel_vec, nhat, xedge, Ql, Qr, Auxl, Auxr, Fl, Fr);
            smax.set(i, Max(smax_edge,smax.get(i)) );

            // Construct fluxes
            for (int m=1; m<=meqn; m++)
            {
                Fr_tmp.set(m,ell, Fr.get(m) );
                Fl_tmp.set(m,ell, Fl.get(m) );
            }
        }

        // Add edge integral to line integral around the full element
        for (int m=1; m<=meqn; m++)
        for (int k=1; k<=kmax; k++)
        {
            double Fl_sum = 0.0;
            double Fr_sum = 0.0;
            for (int ell=1; ell<=dogParams.get_space_order(); ell++)
            {
                Fl_sum = Fl_sum + 0.5*EdgeData.wgts1d->get(ell)
                    *EdgeData.phi_left->get(i,ell,k) *Fl_tmp.get(m,ell);
                Fr_sum = Fr_sum + 0.5*EdgeData.wgts1d->get(ell)
                    *EdgeData.phi_right->get(i,ell,k)*Fr_tmp.get(m,ell);
            }
            EdgeFluxIntegral.set(ileft, m,k, EdgeFluxIntegral.get(ileft, m,k) + Fl_sum/Areal );
            EdgeFluxIntegral.set(iright,m,k, EdgeFluxIntegral.get(iright,m,k) - Fr_sum/Arear );
        }
    }
    // ---------------------------------------------------------

    // ---------------------------------------------------------
    // Part II: compute intra-element contributions
    // ---------------------------------------------------------
    L2ProjectGrad_Unst(vel_vec, 1,NumPhysElems,
            space_order,space_order,space_order,space_order,
            Mesh,&q,&aux,&ElemFluxIntegral,&FluxFunc);
    // ---------------------------------------------------------

    // ---------------------------------------------------------
    // Part III: compute source term
    // --------------------------------------------------------- 
    if ( dogParams.get_source_term()>0 )
    {        
        // Set source term on computational grid
        // Set values and apply L2-projection
        L2Project_Unst(t, vel_vec, 1,NumPhysElems,
                space_order,space_order,space_order,space_order,
                Mesh,&q,&aux,&Psi,&SourceTermFunc);
    }
    // ---------------------------------------------------------

    // ---------------------------------------------------------
    // Part IV: construct Lstar
    // ---------------------------------------------------------
    if (dogParams.get_source_term()==0)  // Without Source Term
    { 
#pragma omp parallel for
        for (int i=1; i<=NumPhysElems; i++)	
        for (int m=1; m<=meqn; m++)
        for (int k=1; k<=kmax; k++)
        {
            double tmp = ElemFluxIntegral.get(i,m,k) - EdgeFluxIntegral.get(i,m,k);
            Lstar.set(i,m,k, tmp );	      
        }
    }
    else  // With Source Term
    {
#pragma omp parallel for
        for (int i=1; i<=NumPhysElems; i++)
        for (int m=1; m<=meqn; m++)
        for (int k=1; k<=kmax; k++)
        {
            double tmp = ElemFluxIntegral.get(i,m,k) 
                - EdgeFluxIntegral.get(i,m,k)
                + Psi.get(i,m,k);

            Lstar.set(i,m,k, tmp );
        }
    }
    // ---------------------------------------------------------

    // ---------------------------------------------------------
    // Part V: add extra contributions to Lstar
    // ---------------------------------------------------------
    // Call LstarExtra
    LstarExtra_Unst(Mesh,&q,&aux,&Lstar);
    // ---------------------------------------------------------

}