// 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 );
}
}
}
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");
}
// 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); }
// ---------------------------------------------------------
}
// This file should be identical to DogSolveRK_Unst, with the exception that all
// output printing statements are silenced.
//
// Advance the solution qold to qnew over time interval tstart to tend.
//
// All local information is allocated within this function. The only part
// that gets shared are time values passed through dogStateUnst2. This class
// should be modified to accept the state variable, q and aux in place of only
// containing time information as is currently the case. (-DS)
double DogSolveRK_Unst_Quiet(
const dTensor2* vel_vec,
const mesh& Mesh, const edge_data_Unst& EdgeData,
dTensor3& aux, dTensor3& qold, dTensor3& qnew,
const double tstart, const double tend, DogStateUnst2& dogStateUnst2)
{
const int mx = qnew.getsize(1);
const int meqn = qnew.getsize(2);
const int kmax = qnew.getsize(3);
const int maux = aux.getsize(2);
const double* cflv = dogParams.get_cflv();
const int nv = dogParams.get_nv();
RKinfo rk;
SetRKinfo(dogParams.get_time_order(),rk);
// define local variables
int n_step = 0;
double t = tstart;
double dt = dogStateUnst2.get_initial_dt();
const double CFL_max = cflv[1];
const double CFL_target = cflv[2];
double cfl = 0.0;
double dtmin = dt;
double dtmax = dt;
const int NumElems = Mesh.get_NumElems(); // Number of total elements in mesh
const int NumNodes = Mesh.get_NumNodes(); // Number of nodes in mesh
const int NumEdges = Mesh.get_NumEdges(); // Number of edges in mesh
dTensor3 qstar(NumElems,meqn,kmax);
dTensor3 q1(NumElems,meqn,kmax);
dTensor3 q2(NumElems,meqn,kmax);
dTensor3 auxstar(NumElems,maux,kmax);
dTensor3 Lstar(NumElems,meqn,kmax);
dTensor3 Lold(NumElems,meqn,kmax);
dTensor3 auxold(NumElems,maux,kmax);
dTensor1 smax(NumEdges);
void L2Project_Unst(
const dTensor2* vel_vec,
const int istart, const int iend,
const int QuadOrder, const int BasisOrder_qin, const int BasisOrder_auxin, const
int BasisOrder_fout, const mesh& Mesh, const dTensor3* qin, const dTensor3*
auxin, dTensor3* fout, void (*Func)(const dTensor2* vel_vec, const
dTensor2&,const dTensor2&, const dTensor2&,dTensor2&));
// JUNK here:
void AuxFuncWrapper(
const dTensor2* vel_vec,
const dTensor2& xpts,
const dTensor2& NOT_USED_1,
const dTensor2& NOT_USED_2,
dTensor2& auxvals);
const int space_order = dogParams.get_space_order();
if( maux > 0 )
{
printf("WARNING: maux = %d should be zero for Vlasov routines.", maux);
printf(" Modify parameters.ini to remove this warning\n" );
L2Project_Unst(vel_vec,1,NumElems,
space_order,space_order,space_order,space_order,
Mesh,&qnew,&aux,&aux,&AuxFuncWrapper);
}
// Set initialize qstar and auxstar values
qstar.copyfrom(qold);
auxstar.copyfrom(aux);
// Runge-Kutta time stepping
while (t<tend)
{
// initialize time step
int m_accept = 0;
n_step = n_step + 1;
// check if max number of time steps exceeded
if( n_step > nv )
{
eprintf(" Error in DogSolveRK_Unst.cpp: "
" Exceeded allowed # of time steps \n"
" n_step = %d\n"
" nv = %d\n\n",
n_step, nv);
}
// copy qnew into qold
qold.copyfrom(qnew);
auxold.copyfrom(aux);
// keep trying until we get a dt that does not violate CFL condition
while (m_accept==0)
{
// set current time
double told = t;
if (told+dt > tend)
{ dt = tend - told; }
t = told + dt;
// TODO - this needs to be performed at the 'local' level
dogStateUnst2.set_time ( told );
dogStateUnst2.set_dt ( dt );
// Set initial maximum wave speed to zero
smax.setall(0.);
// Take a full time step of size dt
switch ( dogParams.get_time_order() )
{
case 1: // First order in time (1-stage)
// -----------------------------------------------
// Stage #1 (the only one in this case)
rk.mstage = 1;
BeforeStep_Unst(dt,Mesh,aux,qnew);
ConstructL_Unst(told, vel_vec,Mesh,EdgeData,aux,qnew,Lstar,smax);
UpdateSoln_Unst(rk.alpha1->get(rk.mstage),rk.alpha2->get(rk.mstage),
rk.beta->get(rk.mstage),dt,Mesh,aux, qnew, Lstar, qnew);
AfterStep_Unst(dt,Mesh,aux,qnew);
// -----------------------------------------------
break;
case 2: // Second order in time (2-stages)
// -----------------------------------------------
// Stage #1
rk.mstage = 1;
dogStateUnst2.set_time(told);
BeforeStep_Unst(dt,Mesh,aux,qnew);
ConstructL_Unst(told,vel_vec,Mesh,EdgeData,aux,qnew,Lstar,smax);
UpdateSoln_Unst(
rk.alpha1->get(rk.mstage),rk.alpha2->get(rk.mstage),
rk.beta->get(rk.mstage), dt, Mesh, aux, qnew, Lstar, qstar);
AfterStep_Unst(dt, Mesh, auxstar, qstar);
// ------------------------------------------------
// Stage #2
rk.mstage = 2;
dogStateUnst2.set_time(told+dt);
BeforeStep_Unst(dt, Mesh, auxstar, qstar);
ConstructL_Unst(told+1.0*dt, vel_vec, Mesh, EdgeData, aux, qstar, Lstar, smax);
UpdateSoln_Unst(rk.alpha1->get(rk.mstage), rk.alpha2->get(rk.mstage),
rk.beta->get(rk.mstage), dt, Mesh, auxstar, qstar, Lstar, qnew);
AfterStep_Unst(dt, Mesh, aux, qnew);
// ------------------------------------------------
break;
case 3: // Third order in time (3-stages)
// qnew = alpha1 * qstar + alpha2 * qnew + beta * dt * L( qstar )
// alpha1 = 1.0
// alpha2 = 0.0
// beta = 1.0
// ------------------------------------------------
// Stage #1
rk.mstage = 1;
dogStateUnst2.set_time(told);
BeforeStep_Unst(dt,Mesh,aux,qnew);
ConstructL_Unst(told, vel_vec,Mesh,EdgeData,aux,qnew,Lstar,smax);
Lold.copyfrom(Lstar);
UpdateSoln_Unst(rk.alpha1->get(rk.mstage),rk.alpha2->get(rk.mstage),
rk.beta->get(rk.mstage),dt,Mesh,aux,qnew,Lstar,qstar);
AfterStep_Unst(dt,Mesh,aux,qstar);
// -------------------------------------------------
// alpha1 = 0.75
// alpha2 = 0.25
// beta = 0.25
// Stage #2
rk.mstage = 2;
dogStateUnst2.set_time(told+0.5*dt);
BeforeStep_Unst(dt,Mesh,aux,qstar);
ConstructL_Unst(told+dt, vel_vec,Mesh,EdgeData,aux,qstar,Lstar,smax);
UpdateSoln_Unst(rk.alpha1->get(rk.mstage),rk.alpha2->get(rk.mstage),
rk.beta->get(rk.mstage),dt,Mesh,aux,qnew,Lstar,qstar);
AfterStep_Unst(dt,Mesh,aux,qstar);
// --------------------------------------------------
// alpha1 = 2/3
// alpha2 = 1/3
// beta = 2/3
// Stage #3
rk.mstage = 3;
dogStateUnst2.set_time(told+dt);
BeforeStep_Unst(dt,Mesh,auxstar,qstar);
ConstructL_Unst(told+0.5*dt,vel_vec,Mesh,EdgeData,auxstar,qstar,Lstar,smax);
UpdateSoln_Unst(rk.alpha1->get(rk.mstage),rk.alpha2->get(rk.mstage),
rk.beta->get(rk.mstage),dt,Mesh,aux,qstar,Lstar,qnew);
AfterStep_Unst(dt,Mesh,aux,qnew);
// --------------------------------------------------
break;
default: unsupported_value_error(dogParams.get_time_order());
}
// compute cfl number
cfl = GetCFL_Unst(dt,Mesh,aux,smax);
// output time step information
// if (dogParams.get_verbosity()>0)
// {
// printf(" In DogSolveRK_Quiet: DogSolve2D ... Step %5d"
// " CFL =%6.3f"
// " dt =%11.3e"
// " t =%11.3e\n",
// n_step, cfl, dt, t);
// }
// choose new time step
if (cfl>0.0)
{
dt = Min(dogParams.get_max_dt(), dt*CFL_target/cfl);
dtmin = Min(dt,dtmin);
dtmax = Max(dt,dtmax);
}
else
{
dt = dogParams.get_max_dt();
}
// see whether to accept or reject this step
if (cfl<=CFL_max)
// accept
{
m_accept = 1;
dogStateUnst2.set_time(t);
// do any extra work
// AfterFullTimeStep_Unst(dogStateUnst2.get_dt(),Mesh,
// auxold,qold,Lold,aux,qnew);
}
else
//reject
{
t = told;
dogStateUnst2.set_time(told);
// if( dogParams.get_verbosity() > 0 )
// {
// printf("DogSolve2D rejecting step..."
// "CFL number too large\n");
// }
// copy qold into qnew
qnew.copyfrom(qold);
aux.copyfrom(auxold);
// after reject function
// AfterReject_Unst(Mesh,dt,aux,qnew);
}
}
}
// printf(" Finished! t = %2.3e and nsteps = %d\n", t, n_step );
// set initial time step for next call to DogSolveRK
dogStateUnst2.set_initial_dt(dt);
void DeleteRKInfo(RKinfo& rk);
DeleteRKInfo(rk);
return cfl;
}
// 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 );
}
}
}
// 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 );
}
}
}
}
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);
// ---------------------------------------------------------
}
