// This is a user-supplied routine that sets the the boundary conditions
//
void SetBndValues_Unst(const mesh& Mesh, dTensor3* q, dTensor3* aux)
{
int meqn = q->getsize(2);
int kmax = q->getsize(3);
int maux = aux->getsize(2);
int NumElems = Mesh.get_NumElems();
int NumPhysElems = Mesh.get_NumPhysElems();
int NumGhostElems = Mesh.get_NumGhostElems();
int NumNodes = Mesh.get_NumNodes();
int NumPhysNodes = Mesh.get_NumPhysNodes();
int NumEdges = Mesh.get_NumEdges();
// ----------------------------------------
// Loop over each ghost cell element and
// place the correct information into
// these elements
// ----------------------------------------
for (int i=1; i<=NumGhostElems; i++)
{
int j = Mesh.get_ghost_link(i);
for (int m=1; m<=meqn; m++)
for (int k=1; k<=kmax; k++)
{
q->set(i+NumPhysElems,m,k, 0.0 );
}
}
}
void ConSoln_Unst(const mesh& Mesh,
const dTensor3& aux,
const dTensor3& q,
double t,
string outputdir)
{
const int NumElems = q.getsize(1);
const int meqn = q.getsize(2);
const int kmax = q.getsize(3);
const int maux = aux.getsize(2);
const string fname1 = outputdir+"/conservation.dat";
ofstream write_file1,write_file2;
dTensor1 qsum(meqn);
dTensor1 res_sum(meqn);
const int NumPhysElems = Mesh.get_NumPhysElems();
if (t==0)
{
write_file1.open(fname1.c_str(), ofstream::out);
}
else
{
write_file1.open(fname1.c_str(), ofstream::app);
}
// -----------------
// CONSERVATION
// -----------------
if (dogParams.get_mcapa()<1) // without capacity function
{
for (int m=1; m<=meqn; m++)
{
qsum.set(m,0.0);
for (int i=1; i<=NumPhysElems; i++)
{
double dtmp = Mesh.get_area_prim(i);
double qtmp = q.get(i,m,1);
qsum.set(m, (qsum.get(m) + dtmp*qtmp) );
}
}
}
write_file1 << setprecision(16);
write_file1 << setw(24) << scientific << t << " ";
for (int m=1; m<=meqn; m++)
{
if (fabs(qsum.get(m)) < 1.0e-99) {qsum.set(m, 0.0);}
write_file1 << setw(24) << scientific << qsum.get(m) << " ";
}
write_file1 << endl;
write_file1.close();
}
double GetCFL_Unst(double dt, const mesh& Mesh,
const dTensor3& aux, const dTensor1& smax)
{
double cfl=-100.0;
int NumPhysElems = Mesh.get_NumPhysElems();
int NumEdges = Mesh.get_NumEdges();
for (int i=1; i<=NumPhysElems; i++)
{
double Area = Mesh.get_area_prim(i);
int edge1 = Mesh.get_tedge(i,1);
int edge2 = Mesh.get_tedge(i,2);
int edge3 = Mesh.get_tedge(i,3);
double tmp = Max(Max(smax.get(edge1),smax.get(edge2)),
smax.get(edge3));
cfl = Max(0.5*dt*tmp/Area, cfl);
}
return cfl;
}
// This is a user-supplied routine that sets the the boundary conditions
//
// The default routine for the 4D Vlasov code is to apply zero boundary
// conditions in configuration space. This routine is identical to the one
// found in the unst branch of the 2D DoGPack code, and *should* be setting
// periodic boundary conditions.
//
void SetBndValues_Unst(const mesh& Mesh, dTensor3* q, dTensor3* aux)
{
// problem information (If this were to be pulled from DogParams, these
// numbers would NOT be correct! The reason is that each quadrature point
// was actually saved as a separate "equation")
const int meqn = q->getsize(2);
const int kmax = q->getsize(3);
const int maux = aux->getsize(2);
// Mesh information
const int NumElems = Mesh.get_NumElems();
const int NumPhysElems = Mesh.get_NumPhysElems();
const int NumGhostElems = Mesh.get_NumGhostElems();
const int NumNodes = Mesh.get_NumNodes();
const int NumPhysNodes = Mesh.get_NumPhysNodes();
const int NumEdges = Mesh.get_NumEdges();
// ----------------------------------------
// Loop over each ghost cell element and
// place the correct information into
// these elements
// ----------------------------------------
for (int i=1; i<=NumGhostElems; i++)
{
int j = Mesh.get_ghost_link(i);
for (int m=1; m<=meqn; m++)
for (int k=1; k<=kmax; k++)
{
q->set(i+NumPhysElems, m, k, q->get(j, m, k) );
}
}
}
//
// Output basic mesh information to screen
//
void ScreenOutput(const mesh& Mesh)
{
// Compute mesh quality parameters
double totalarea = Mesh.get_area_prim(1);
double maxarea = Mesh.get_area_prim(1);
double minarea = Mesh.get_area_prim(1);
for (int i=2; i<=Mesh.get_NumPhysElems(); i++)
{
double tmp = Mesh.get_area_prim(i);
totalarea = totalarea + tmp;
if (tmp < minarea)
{ minarea = tmp; }
if (tmp > maxarea)
{ maxarea = tmp; }
}
double minAngle = 180.0;
for (int i=1; i<=Mesh.get_NumPhysElems(); i++)
{
const int i1 = Mesh.get_tnode(i,1);
const int i2 = Mesh.get_tnode(i,2);
const int i3 = Mesh.get_tnode(i,3);
point v12, v23, v31;
v12.x = Mesh.get_node(i2,1) - Mesh.get_node(i1,1);
v12.y = Mesh.get_node(i2,2) - Mesh.get_node(i1,2);
v23.x = Mesh.get_node(i3,1) - Mesh.get_node(i2,1);
v23.y = Mesh.get_node(i3,2) - Mesh.get_node(i2,2);
v31.x = Mesh.get_node(i1,1) - Mesh.get_node(i3,1);
v31.y = Mesh.get_node(i1,2) - Mesh.get_node(i3,2);
double angle1 = acos((v12.x*-v31.x+v12.y*-v31.y)
/(sqrt(v12.x*v12.x+v12.y*v12.y)*sqrt(v31.x*v31.x+v31.y*v31.y)));
double angle2 = acos((v23.x*-v12.x+v23.y*-v12.y)
/(sqrt(v23.x*v23.x+v23.y*v23.y)*sqrt(v12.x*v12.x+v12.y*v12.y)));
double angle3 = acos((v31.x*-v23.x+v31.y*-v23.y)
/(sqrt(v31.x*v31.x+v31.y*v31.y)*sqrt(v23.x*v23.x+v23.y*v23.y)));
if ((angle1*180/pi) < minAngle)
{ minAngle = angle1*180/pi; }
if ((angle2*180/pi) < minAngle)
{ minAngle = angle2*180/pi; }
if ((angle3*180/pi) < minAngle)
{ minAngle = angle3*180/pi; }
}
// Output summary of results to screen
printf("\n");
printf(" SUMMARY OF RESULTS:\n");
printf(" -------------------\n");
printf(" Number of Elements: %8i\n",Mesh.get_NumElems());
printf(" Number of Physical Elements: %8i\n",Mesh.get_NumPhysElems());
printf(" Number of Ghost Elements: %8i\n",Mesh.get_NumGhostElems());
printf(" Number of Nodes: %8i\n",Mesh.get_NumNodes());
printf(" Number of Physical Nodes: %8i\n",Mesh.get_NumPhysNodes());
printf(" Number of Boundary Nodes: %8i\n",Mesh.get_NumBndNodes());
printf(" Number of Edges: %8i\n",Mesh.get_NumEdges());
printf(" Number of Boundary Edges: %8i\n",Mesh.get_NumBndEdges());
printf("\n");
printf(" Total Area Covered: %24.16e\n",totalarea);
printf(" Area Ratio: small/large: %24.16e\n",minarea/maxarea);
printf(" Angle Ratio: minAngle/60: %24.16e\n",minAngle/60.0);
printf("\n");
}
void ComputeError(const int space_order,
const mesh& Mesh,
const dTensor1& phi,
const dTensor2& E1,
const dTensor2& E2,
void (*PhiFunc)(const dTensor2& xpts,dTensor2& phi_ex),
void (*EfieldFunc)(const dTensor2& xpts,dTensor2& Efield_ex))
{
// Potential
const int NumPhysNodes = phi.getsize();//Mesh.get_NumPhysNodes();
dTensor2 xpts(NumPhysNodes,2);
dTensor2 phi_ex(NumPhysNodes,1);
double phi_err;
double phi_rel;
switch(space_order)
{
case 2:
for (int i=1; i<=NumPhysNodes; i++)
{
xpts.set(i,1, Mesh.get_node(i,1) );
xpts.set(i,2, Mesh.get_node(i,2) );
}
PhiFunc(xpts,phi_ex);
phi_err = 0.0;
phi_rel = 0.0;
for (int i=1; i<=NumPhysNodes; i++)
{
phi_rel = phi_rel + pow(phi_ex.get(i,1),2);
phi_err = phi_err + pow(phi_ex.get(i,1)-phi.get(i),2);
}
phi_err = sqrt(phi_err/phi_rel);
break;
case 3:
for (int i=1; i<=NumPhysNodes; i++)
{
xpts.set(i,1, Mesh.get_sub_node(i,1) );
xpts.set(i,2, Mesh.get_sub_node(i,2) );
}
PhiFunc(xpts,phi_ex);
phi_err = 0.0;
phi_rel = 0.0;
for (int i=1; i<=NumPhysNodes; i++)
{
phi_rel = phi_rel + pow(phi_ex.get(i,1),2);
phi_err = phi_err + pow(phi_ex.get(i,1)-phi.get(i),2);
}
phi_err = sqrt(phi_err/phi_rel);
break;
}
// Electric field components
void L2Project_Unst(const int istart,
const int iend,
const int QuadOrder,
const int BasisOrder_fout,
const mesh& Mesh,
dTensor3* fout,
void (*Func)(const dTensor2&,dTensor2&));
const int NumElems = Mesh.get_NumElems();
const int NumPhysElems = Mesh.get_NumPhysElems();
const int kmax = E1.getsize(2);
dTensor3 Efield_ex(NumElems,2,kmax);
L2Project_Unst(1,NumElems,space_order,space_order,
Mesh,&Efield_ex,EfieldFunc);
double E1_err = 0.0;
double E1_rel = 0.0;
double E2_err = 0.0;
double E2_rel = 0.0;
for (int i=1; i<=NumPhysElems; i++)
{
double Area = Mesh.get_area_prim(i);
double tmp1 = 0.0;
double tmp2 = 0.0;
double tmp1_rel = 0.0;
double tmp2_rel = 0.0;
for (int k=1; k<=kmax; k++)
{
tmp1 = tmp1 + pow((E1.get(i,k)-Efield_ex.get(i,1,k)),2);
tmp2 = tmp2 + pow((E2.get(i,k)-Efield_ex.get(i,2,k)),2);
tmp1_rel = tmp1_rel + pow((Efield_ex.get(i,1,k)),2);
tmp2_rel = tmp2_rel + pow((Efield_ex.get(i,2,k)),2);
}
E1_err = E1_err + Area*tmp1;
E2_err = E2_err + Area*tmp2;
E1_rel = E1_rel + Area*tmp1_rel;
E2_rel = E2_rel + Area*tmp2_rel;
}
E1_err = sqrt(E1_err/E1_rel);
E2_err = sqrt(E2_err/E2_rel);
// Summary
printf(" |----------------------------\n");
printf(" | Errors:\n");
printf(" |----------------------------\n");
printf(" | phi_err = %e\n",phi_err);
printf(" | E1_err = %e\n",E1_err);
printf(" | E2_err = %e\n",E2_err);
printf(" |----------------------------\n");
printf("\n");
}
// Save the time, and print the L2-norm of the electric field
//
// TODO - is this the spot where we'd like to print moments? Do we need
// moments after each time step or for each frame?
//
void PrintElectricField( const double t, const mesh& Mesh, const dTensorBC5& q,
const dTensor2& E1, const dTensor2& E2 )
{
const int mx = q.getsize(1);
const int my = q.getsize(2);
const int NumElems = q.getsize(3);
const int meqn = q.getsize(4);
const int kmax = q.getsize(5);
const int mbc = q.getmbc();
const int NumPhysElems = Mesh.get_NumPhysElems();
string fname1 = string(dogParams.get_outputdir())+"/conservation.dat";
string fname2 = string(dogParams.get_outputdir())+"/Efield.dat";
ofstream write_file1,write_file2;
if( fabs(t) < EPSILON )
{
write_file1.open(fname1.c_str(), ofstream::out);
write_file2.open(fname2.c_str(), ofstream::out);
}
else
{
write_file1.open(fname1.c_str(), ofstream::app);
write_file2.open(fname2.c_str(), ofstream::app);
}
// -----------------
// CONSERVATION
// -----------------
dTensor1 qsum(meqn);
if (dogParams.get_mcapa()<1) // without capacity function
{
for (int m=1; m<=meqn; m++)
{
qsum.set(m,0.0);
for (int i=1; i<=mx; i++)
for (int j=1; j<=my; j++)
for (int n=1; n<=NumPhysElems; n++)
{
qsum.set(m, qsum.get(m) + Mesh.get_area_prim(n)*dogParamsCart2.get_prim_vol()*q.get(i,j,n,m,1) );
}
}
}
write_file1 << setprecision(16);
write_file1 << setw(24) << scientific << t << " ";
for (int m=1; m<=meqn; m++)
{
if (fabs(qsum.get(m)) < 1.0e-99) {
qsum.set(m, 0.0);
}
write_file1 << setw(24) << scientific << qsum.get(m) << " ";
}
write_file1 << endl;
write_file1.close();
///////////////////////////////////////////////////////////////////////////
// Conserved Vlasov-Poisson Quantities //
// ||f||_1, ||f||_2, Energy, Entropy //
///////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////
// Electric Field
const int mcons = 4;
// TODO - do we want to compute the extra moments here or later?
////dTensorBC4 ConsData2d(mx, my, mcons, kmax,mbc);
////const int space_order = dogParams.get_space_order();
////L2Project(1, mx, 1, my, space_order, space_order, space_order,
//// space_order, &q, &aux, &ConsData2d, &ConservedFunc );
// electric field, entropy and energy
dTensor1 ConsData(mcons);
ConsData.setall(0.);
// Compute the integral of all the 2d quantities:
// for (int m=1; m<=mcons; m++)
// {
// for (int i=1; i<=mx; i++)
// for (int j=1; j<=my; j++)
// {
// ConsData.set(m, ConsData.get(m) + dx*dy*ConsData2d.get(i,j,m,1) );
// }
// }
// Compute the integral of all the 1d quanties:
// \int E^2 \ dx
double Esqd = 0.0;
for(int n=1; n <= NumElems; n++ )
for(int k=1; k <= E1.getsize(2); k++ )
{
Esqd += Mesh.get_area_prim(n) * (
pow( E1.get(n,k), 2 ) + pow(E2.get(n,k), 2 )
);
}
//printf("Esqd = %f\n", Esqd );
//exit(1);
// Add in E to the total Energy:
// ConsData.set(3, ConsData.get(3) + 0.5 * E2 );
ConsData.set(3, 0. );
// print time:
write_file2 << setprecision(16);
write_file2 << setw(24) << scientific << t << " ";
// print ||E||_2 first:
write_file2 << setw(24) << scientific << sqrt( Esqd ) << " ";
// print all the other conserved quantities:
for (int m=1; m<=mcons; m++)
{
write_file2 << setw(24) << scientific << ConsData.get(m) << " ";
}
write_file2 << endl;
write_file2.close();
}
// 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); }
// ---------------------------------------------------------
}
void ConstructA_CG2(const mesh& Mesh, FullMatrix& A)
{
const int NumPhysElems = Mesh.get_NumPhysElems();
const int NumBndNodes = Mesh.get_SubNumBndNodes();
const int Asize = Mesh.get_SubNumPhysNodes();
assert_eq(Asize,A.get_NumRows());
assert_eq(Asize,A.get_NumCols());
dTensor1 A1(6);
dTensor1 A2(6);
dTensor1 A3(6);
dTensor1 A4(6);
dTensor1 A5(6);
dTensor1 A6(6);
A1.set(1, -oneninth );
A1.set(2, 4.0*oneninth );
A1.set(3, -oneninth );
A1.set(4, 4.0*oneninth );
A1.set(5, 4.0*oneninth );
A1.set(6, -oneninth );
A2.set(1, -onethird );
A2.set(2, 0.0 );
A2.set(3, onethird );
A2.set(4, -4.0*onethird );
A2.set(5, 4.0*onethird );
A2.set(6, 0.0 );
A3.set(1, -onethird );
A3.set(2, -4.0*onethird );
A3.set(3, 0.0 );
A3.set(4, 0.0 );
A3.set(5, 4.0*onethird );
A3.set(6, onethird );
A4.set(1, 4.0 );
A4.set(2, -4.0 );
A4.set(3, 0.0 );
A4.set(4, -4.0 );
A4.set(5, 4.0 );
A4.set(6, 0.0 );
A5.set(1, 2.0 );
A5.set(2, -4.0 );
A5.set(3, 2.0 );
A5.set(4, 0.0 );
A5.set(5, 0.0 );
A5.set(6, 0.0 );
A6.set(1, 2.0 );
A6.set(2, 0.0 );
A6.set(3, 0.0 );
A6.set(4, -4.0 );
A6.set(5, 0.0 );
A6.set(6, 2.0 );
dTensor2 spts(3,2);
spts.set(1,1, 1.0/3.0 );
spts.set(1,2, -1.0/6.0 );
spts.set(2,1, -1.0/6.0 );
spts.set(2,2, -1.0/6.0 );
spts.set(3,1, -1.0/6.0 );
spts.set(3,2, 1.0/3.0 );
dTensor1 wgts(3);
wgts.set(1, 1.0/6.0 );
wgts.set(2, 1.0/6.0 );
wgts.set(3, 1.0/6.0 );
// Loop over all elements in the mesh
for (int i=1; i<=NumPhysElems; i++)
{
// Information for element i
iTensor1 tt(6);
for (int k=1; k<=6; k++)
{ tt.set(k, Mesh.get_node_subs(i,k) ); }
// Evaluate gradients of the Lagrange polynomials on Gauss quadrature points
dTensor2 gpx(6,3);
dTensor2 gpy(6,3);
for (int m=1; m<=3; m++)
{
double xi = spts.get(m,1);
double eta = spts.get(m,2);
for (int k=1; k<=6; k++)
{
double gp_xi = A2.get(k) + 2.0*A5.get(k)*xi + A4.get(k)*eta;
double gp_eta = A3.get(k) + A4.get(k)*xi + 2.0*A6.get(k)*eta;
gpx.set(k,m, Mesh.get_jmat(i,1,1)*gp_xi
+ Mesh.get_jmat(i,1,2)*gp_eta );
gpy.set(k,m, Mesh.get_jmat(i,2,1)*gp_xi
+ Mesh.get_jmat(i,2,2)*gp_eta );
}
}
// Entries of the stiffness matrix A
double Area = Mesh.get_area_prim(i);
for (int j=1; j<=6; j++)
for (int k=1; k<=6; k++)
{
double tmp = A.get(tt.get(j),tt.get(k));
for (int m=1; m<=3; m++)
{
tmp = tmp + 2.0*Area*wgts.get(m)*(gpx.get(j,m)*gpx.get(k,m)+gpy.get(j,m)*gpy.get(k,m));
}
A.set(tt.get(j),tt.get(k), tmp );
}
}
// Replace boundary node equations by Dirichlet boundary condition enforcement
for (int i=1; i<=NumBndNodes; i++)
{
const int j=Mesh.get_sub_bnd_node(i);
for (int k=1; k<=A.get_NumCols(); k++)
{
A.set(j,k, 0.0 );
}
for (int k=1; k<=A.get_NumRows(); k++)
{
A.set(k,j, 0.0 );
}
A.set(j,j, 1.0 );
}
// Get sparse structure representation
A.Sparsify();
}
// This is the positivity preserving limiter proposed in
// "Maximum-Principle-Satisfying and Positivity-Preserving
// High Order Discontinuous Galerkin Schemes
// for Conservation Laws on Triangular Meshes", Zhang, Xia and Shu
// J. Sci. Comput. (2012).
//
// THIS METHOD ASSUMES THAT EVERY COMPONENT OF CONSERVED VARIABLES SHOULD STAY
// POSITIVE.
//
// In order to implement this for a different scheme, one should rewrite, or
// redefine what components should remain positiive. This will require
// reworking the control flow logic for how time step lengths are chosen.
void ApplyPosLimiter_Unst(const mesh& Mesh, const dTensor3& aux, dTensor3& q)
{
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();
// Do nothing in the case of piecewise constants
if( space_order == 1 )
{ return; }
// ------------------------------------------------ //
// number of points where we want to check solution //
// ------------------------------------------------ //
const int space_order_sq = space_order*space_order;
const int mpts_vec[] = {0, 3*space_order_sq, 18, 3*space_order_sq, 3*space_order_sq }; // TODO - FILL IN 2ND-ORDER CASE
const int mpoints = mpts_vec[space_order-1];
// ---------------------------------------------------------- //
// sample basis at all points where we want to check solution //
// ---------------------------------------------------------- //
dTensor2 spts(mpoints, 2);
void SetPositivePoints_Unst(const int& space_order, dTensor2& spts);
SetPositivePoints_Unst(space_order, spts);
void SamplePhiAtPositivePoints_Unst(const int& space_order,
const dTensor2& spts, dTensor2& phi);
dTensor2 phi(mpoints, kmax);
SamplePhiAtPositivePoints_Unst(space_order, spts, phi);
// -------------------------------------------------------------- //
// q_limited = Q1 + \theta ( q(xi,eta) - Q1 ) //
// where theta = min(1, |Q1| / |Q1-m|; m = min_{i} q(xi_i, eta_i) //
// -------------------------------------------------------------- //
#pragma omp parallel for
for(int i=1; i <= NumPhysElems; i++)
for(int me=1; me <= meqn; me++)
{
double m = 0.0;
for(int mp=1; mp <= mpoints; mp++)
{
// evaluate q at spts(mp) //
double qnow = 0.0;
for( int k=1; k <= kmax; k++ )
{
qnow += q.get(i,me,k) * phi.get(mp,k);
}
m = Min(m, qnow);
}
double theta = 0.0;
double Q1 = q.get(i,me,1); assert_ge( Q1, -1e-13 );
if( fabs( Q1 - m ) < 1.0e-14 ){ theta = 1.0; }
else{ theta = Min( 1.0, fabs( Q1 / (Q1 - m) ) ); }
// limit q //
for( int k=2; k <= kmax; k++ )
{
q.set(i,me,k, q.get(i,me,k) * theta );
}
}
}
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);
// ---------------------------------------------------------
}