#include <iostream>
#include <sstream>
#include <ctime>
#include <cstdlib>
#include <cstdio>
#include <cstring>
#include <cmath>
#include "moab/Core.hpp"
#include "moab/Interface.hpp"
#include "moab/IntxMesh/Intx2MeshOnSphere.hpp"
#include "moab/ProgOptions.hpp"
#include "MBTagConventions.hpp"
#include "moab/ParallelComm.hpp"
#include "moab/IntxMesh/IntxUtils.hpp"
#include "IntxUtilsCSLAM.hpp"
#include "TestUtil.hpp"

Include dependency graph for diffusion.cpp:

Functions
ErrorCode	add_field_value (Interface *mb, EntityHandle euler_set, int rank, Tag &tagTracer, Tag &tagElem, Tag &tagArea)
ErrorCode	compute_velocity_case1 (Interface *mb, EntityHandle euler_set, Tag &tagh, int rank, int tStep)
ErrorCode	compute_tracer_case1 (Interface *mb, Intx2MeshOnSphere &worker, EntityHandle euler_set, EntityHandle lagr_set, EntityHandle out_set, Tag &tagElem, Tag &tagArea, int rank, int tStep, Range &connecVerts)
int	main (int argc, char **argv)
Variables
const char	BRIEF_DESC [] = "Simulate a transport problem in a semi-Lagrangian formulation\n"
std::ostringstream	LONG_DESC
double	gtol = 1.e-9
double	radius = 1.
int	numSteps = 3
double	T = 1
int	case_number = 1
bool	writeFiles = false
bool	parallelWrite = false
bool	velocity = false
int	field_type = 1

Function Documentation

ErrorCode add_field_value	(	Interface *	mb,
		EntityHandle	euler_set,
		int	rank,
		Tag &	tagTracer,
		Tag &	tagElem,
		Tag &	tagArea
	)

Definition at line 61 of file diffusion.cpp.

References moab::IntxAreaUtils::area_spherical_polygon(), moab::Range::begin(), moab::IntxUtils::cart_to_spherical(), moab::Range::end(), ErrorCode, field_type, moab::Interface::get_connectivity(), moab::Interface::get_coords(), moab::Interface::get_entities_by_dimension(), moab::IntxUtils::SphereCoords::lat, moab::IntxUtils::SphereCoords::lon, MB_CHK_ERR, MB_SUCCESS, MPI_COMM_WORLD, IntxUtilsCSLAM::quasi_smooth_field(), moab::IntxUtils::SphereCoords::R, radius, moab::Range::size(), IntxUtilsCSLAM::slotted_cylinder_field(), IntxUtilsCSLAM::smooth_field(), moab::IntxUtils::spherical_to_cart(), moab::Interface::tag_get_data(), and moab::Interface::tag_iterate().

{
    /*
     * get all plys first, then vertices, then move them on the surface of the sphere
     *  radius is 1., most of the time
     *
     */
    Range polygons;
    ErrorCode rval = mb->get_entities_by_dimension( euler_set, 2, polygons );
    if( MB_SUCCESS != rval ) return rval;

    Range connecVerts;
    rval = mb->get_connectivity( polygons, connecVerts );
    if( MB_SUCCESS != rval ) return rval;

    void* data;  // pointer to the LOC in memory, for each vertex
    int count;

    rval = mb->tag_iterate( tagTracer, connecVerts.begin(), connecVerts.end(), count, data );MB_CHK_ERR( rval );
    // here we are checking contiguity
    assert( count == (int)connecVerts.size() );
    double* ptr_DP = (double*)data;
    // lambda is for longitude, theta for latitude
    // param will be: (la1, te1), (la2, te2), b, c; hmax=1, r=1/2
    // nondivergent flow, page 5, case 1, (la1, te1) = (M_PI, M_PI/3)
    //                                    (la2, te2) = (M_PI, -M_PI/3)
    //                 la1,    te1    la2    te2     b     c  hmax  r
    if( field_type == 1 )  // quasi smooth
    {
        double params[] = { M_PI, M_PI / 3, M_PI, -M_PI / 3, 0.1, 0.9, 1., 0.5 };
        for( Range::iterator vit = connecVerts.begin(); vit != connecVerts.end(); ++vit )
        {
            EntityHandle oldV = *vit;
            CartVect posi;
            rval = mb->get_coords( &oldV, 1, &( posi[0] ) );MB_CHK_ERR( rval );

            moab::IntxUtils::SphereCoords sphCoord = moab::IntxUtils::cart_to_spherical( posi );

            ptr_DP[0] = IntxUtilsCSLAM::quasi_smooth_field( sphCoord.lon, sphCoord.lat, params );
            ;

            ptr_DP++;  // increment to the next node
        }
    }
    else if( 2 == field_type )  // smooth
    {
        CartVect p1, p2;
        moab::IntxUtils::SphereCoords spr;
        spr.R   = 1;
        spr.lat = M_PI / 3;
        spr.lon = M_PI;
        p1      = moab::IntxUtils::spherical_to_cart( spr );
        spr.lat = -M_PI / 3;
        p2      = moab::IntxUtils::spherical_to_cart( spr );
        //                  x1,    y1,     z1,    x2,   y2,    z2,   h_max, b0
        double params[] = { p1[0], p1[1], p1[2], p2[0], p2[1], p2[2], 1, 5. };
        for( Range::iterator vit = connecVerts.begin(); vit != connecVerts.end(); ++vit )
        {
            EntityHandle oldV = *vit;
            CartVect posi;
            rval = mb->get_coords( &oldV, 1, &( posi[0] ) );MB_CHK_ERR( rval );

            moab::IntxUtils::SphereCoords sphCoord = moab::IntxUtils::cart_to_spherical( posi );

            ptr_DP[0] = IntxUtilsCSLAM::smooth_field( sphCoord.lon, sphCoord.lat, params );
            ;

            ptr_DP++;  // increment to the next node
        }
    }
    else if( 3 == field_type )  // slotted
    {
        //                   la1, te1,   la2, te2,       b,   c,   r
        double params[] = { M_PI, M_PI / 3, M_PI, -M_PI / 3, 0.1, 0.9, 0.5 };  // no h_max
        for( Range::iterator vit = connecVerts.begin(); vit != connecVerts.end(); ++vit )
        {
            EntityHandle oldV = *vit;
            CartVect posi;
            rval = mb->get_coords( &oldV, 1, &( posi[0] ) );MB_CHK_ERR( rval );

            moab::IntxUtils::SphereCoords sphCoord = moab::IntxUtils::cart_to_spherical( posi );

            ptr_DP[0] = IntxUtilsCSLAM::slotted_cylinder_field( sphCoord.lon, sphCoord.lat, params );
            ;

            ptr_DP++;  // increment to the next node
        }
    }

    // add average value for quad/polygon (average corners)
    // do some averages

    Range::iterator iter = polygons.begin();
    double local_mass    = 0.;  // this is total mass on one proc
    while( iter != polygons.end() )
    {
        rval = mb->tag_iterate( tagElem, iter, polygons.end(), count, data );MB_CHK_ERR( rval );
        double* ptr = (double*)data;

        rval = mb->tag_iterate( tagArea, iter, polygons.end(), count, data );MB_CHK_ERR( rval );
        double* ptrArea = (double*)data;
        for( int i = 0; i < count; i++, ++iter, ptr++, ptrArea++ )
        {
            const moab::EntityHandle* conn = NULL;
            int num_nodes                  = 0;
            rval                           = mb->get_connectivity( *iter, conn, num_nodes );MB_CHK_ERR( rval );
            if( num_nodes == 0 ) return MB_FAILURE;
            std::vector< double > nodeVals( num_nodes );
            double average = 0.;
            rval           = mb->tag_get_data( tagTracer, conn, num_nodes, &nodeVals[0] );MB_CHK_ERR( rval );
            for( int j = 0; j < num_nodes; j++ )
                average += nodeVals[j];
            average /= num_nodes;
            *ptr = average;

            // now get area
            std::vector< double > coords;
            coords.resize( 3 * num_nodes );
            rval = mb->get_coords( conn, num_nodes, &coords[0] );MB_CHK_ERR( rval );

            moab::IntxAreaUtils sphAreaUtils;
            *ptrArea = sphAreaUtils.area_spherical_polygon( &coords[0], num_nodes, radius );

            // we should have used some
            // total mass:
            local_mass += *ptrArea * average;
        }
    }
    double total_mass = 0.;
    int mpi_err       = MPI_Reduce( &local_mass, &total_mass, 1, MPI_DOUBLE, MPI_SUM, 0, MPI_COMM_WORLD );
    if( MPI_SUCCESS != mpi_err ) return MB_FAILURE;

    if( rank == 0 ) std::cout << "initial total mass:" << total_mass << "\n";

    // now we can delete the tags? not yet
    return MB_SUCCESS;
}

ErrorCode compute_tracer_case1	(	Interface *	mb,
		Intx2MeshOnSphere &	worker,
		EntityHandle	euler_set,
		EntityHandle	lagr_set,
		EntityHandle	out_set,
		Tag &	tagElem,
		Tag &	tagArea,
		int	rank,
		int	tStep,
		Range &	connecVerts
	)

Definition at line 242 of file diffusion.cpp.

References moab::Range::begin(), moab::Interface::clear_meshset(), CORRTAGNAME, moab::Intx2Mesh::create_departure_mesh_3rd_alg(), moab::Interface::delete_entities(), delta_t, IntxUtilsCSLAM::departure_point_case1(), moab::dum, moab::Range::end(), ErrorCode, moab::Interface::get_connectivity(), moab::Interface::get_coords(), moab::Interface::get_entities_by_dimension(), moab::Intx2Mesh::intersect_meshes(), MB_CHK_ERR, MB_TAG_DENSE, MB_TYPE_HANDLE, numSteps, parallelWrite, radius, moab::Interface::set_coords(), moab::subtract(), t, T, moab::Interface::tag_get_data(), moab::Interface::tag_get_handle(), moab::Intx2MeshOnSphere::update_tracer_data(), moab::Interface::write_file(), and writeFiles.

Referenced by main().

{
    ErrorCode rval;
    EntityHandle dum = 0;
    Tag corrTag;
    rval = mb->tag_get_handle( CORRTAGNAME, 1, MB_TYPE_HANDLE, corrTag, MB_TAG_DENSE, &dum );MB_CHK_ERR( rval );

    double t       = tStep * T / numSteps;  // numSteps is global; so is T
    double delta_t = T / numSteps;          // this is global too, actually
    Range polys;
    rval = mb->get_entities_by_dimension( euler_set, 2, polys );MB_CHK_ERR( rval );

    // change coordinates of lagr mesh vertices
    for( Range::iterator vit = connecVerts.begin(); vit != connecVerts.end(); ++vit )
    {
        EntityHandle oldV = *vit;
        CartVect posi;
        rval = mb->get_coords( &oldV, 1, &( posi[0] ) );MB_CHK_ERR( rval );
        // Intx utils, case 1
        CartVect newPos;
        IntxUtilsCSLAM::departure_point_case1( posi, t, delta_t, newPos );
        newPos = radius * newPos;  // do we need this? the radius should be 1
        EntityHandle new_vert;
        rval = mb->tag_get_data( corrTag, &oldV, 1, &new_vert );MB_CHK_ERR( rval );
        // set the new position for the new vertex
        rval = mb->set_coords( &new_vert, 1, &( newPos[0] ) );MB_CHK_ERR( rval );
    }

    // if in parallel, we have to move some elements to another proc, and receive other cells
    // from other procs
    // lagr and euler are preserved
    EntityHandle covering_set;
    rval = worker.create_departure_mesh_3rd_alg( lagr_set, covering_set );MB_CHK_ERR( rval );
    if( writeFiles )  // so if write
    {
        std::stringstream departureMesh;
        departureMesh << "Departure" << rank << "_" << tStep << ".vtk";
        rval = mb->write_file( departureMesh.str().c_str(), 0, 0, &lagr_set, 1 );MB_CHK_ERR( rval );

        std::stringstream newTracer;
        newTracer << "Tracer" << rank << "_" << tStep << ".vtk";
        rval = mb->write_file( newTracer.str().c_str(), 0, 0, &euler_set, 1 );MB_CHK_ERR( rval );

        std::stringstream lagr_cover;
        lagr_cover << "Cover" << rank << "_" << tStep << ".vtk";
        rval = mb->write_file( lagr_cover.str().c_str(), 0, 0, &covering_set, 1 );MB_CHK_ERR( rval );
    }
    // so we have now the departure at the previous time
    // intersect the 2 meshes (what about some checking of convexity?) for sufficient
    // small dt, it is not an issue;

    rval = worker.intersect_meshes( covering_set, euler_set, out_set );MB_CHK_ERR( rval );
    if( writeFiles )  // so if write
    {
        std::stringstream intx_mesh;
        intx_mesh << "Intx" << rank << "_" << tStep << ".vtk";
        rval = mb->write_file( intx_mesh.str().c_str(), 0, 0, &out_set, 1 );MB_CHK_ERR( rval );
    }

    // serially: lagr is the same order as euler;
    // we need to update now the tracer information on each element, based on
    // initial value and areas of each resulting polygons
    if( parallelWrite && tStep == 1 )
    {
        std::stringstream resTrace;
        resTrace << "Tracer"
                 << "_" << tStep - 1 << ".h5m";
        rval = mb->write_file( resTrace.str().c_str(), 0, "PARALLEL=WRITE_PART", &euler_set, 1, &tagElem, 1 );MB_CHK_ERR( rval );
    }
    rval = worker.update_tracer_data( out_set, tagElem, tagArea );MB_CHK_ERR( rval );

    if( parallelWrite )
    {
        std::stringstream resTrace;
        resTrace << "Tracer"
                 << "_" << tStep << ".h5m";
        rval = mb->write_file( resTrace.str().c_str(), 0, "PARALLEL=WRITE_PART", &euler_set, 1, &tagElem, 1 );
    }

    if( writeFiles )  // so if write
    {
        std::stringstream newIntx;
        newIntx << "newIntx" << rank << "_" << tStep << ".vtk";
        rval = mb->write_file( newIntx.str().c_str(), 0, 0, &out_set, 1 );MB_CHK_ERR( rval );
    }
    // delete now the polygons and the elements of out_set
    // also, all verts that are not in euler set or lagr_set
    Range allVerts;
    rval = mb->get_entities_by_dimension( 0, 0, allVerts );MB_CHK_ERR( rval );

    Range allElems;
    rval = mb->get_entities_by_dimension( 0, 2, allElems );MB_CHK_ERR( rval );

    // add to polys range the lagr polys
    // do not delete lagr set either, with its vertices
    rval = mb->get_entities_by_dimension( lagr_set, 2, polys );MB_CHK_ERR( rval );
    // add to the connecVerts range all verts, from all initial polys
    Range vertsToStay;
    rval = mb->get_connectivity( polys, vertsToStay );MB_CHK_ERR( rval );

    Range todeleteVerts = subtract( allVerts, vertsToStay );

    Range todeleteElem = subtract( allElems, polys );

    // empty the out mesh set
    rval = mb->clear_meshset( &out_set, 1 );MB_CHK_ERR( rval );

    rval = mb->delete_entities( todeleteElem );MB_CHK_ERR( rval );
    rval = mb->delete_entities( todeleteVerts );MB_CHK_ERR( rval );
    if( rank == 0 ) std::cout << " step: " << tStep << "\n";
    return rval;
}

ErrorCode compute_velocity_case1	(	Interface *	mb,
		EntityHandle	euler_set,
		Tag &	tagh,
		int	rank,
		int	tStep
	)

Definition at line 199 of file diffusion.cpp.

References moab::Range::begin(), moab::Range::end(), ErrorCode, moab::Interface::get_connectivity(), moab::Interface::get_coords(), moab::Interface::get_entities_by_dimension(), MB_CHK_ERR, MB_SUCCESS, numSteps, moab::Range::size(), t, T, moab::Interface::tag_iterate(), IntxUtilsCSLAM::velocity_case1(), and moab::Interface::write_file().

Referenced by main().

{
    Range polygons;
    ErrorCode rval = mb->get_entities_by_dimension( euler_set, 2, polygons );
    if( MB_SUCCESS != rval ) return rval;

    Range connecVerts;
    rval = mb->get_connectivity( polygons, connecVerts );
    if( MB_SUCCESS != rval ) return rval;

    void* data;  // pointer to the velo in memory, for each vertex
    int count;

    rval = mb->tag_iterate( tagh, connecVerts.begin(), connecVerts.end(), count, data );MB_CHK_ERR( rval );
    // here we are checking contiguity
    assert( count == (int)connecVerts.size() );
    double* ptr_velo = (double*)data;
    // lambda is for longitude, theta for latitude

    for( Range::iterator vit = connecVerts.begin(); vit != connecVerts.end(); ++vit )
    {
        EntityHandle oldV = *vit;
        CartVect posi;
        rval = mb->get_coords( &oldV, 1, &( posi[0] ) );MB_CHK_ERR( rval );
        CartVect velo;
        double t = T * tStep / numSteps;  //
        IntxUtilsCSLAM::velocity_case1( posi, t, velo );

        ptr_velo[0] = velo[0];
        ptr_velo[1] = velo[1];
        ptr_velo[2] = velo[2];

        // increment to the next node
        ptr_velo += 3;  // to next velocity
    }

    std::stringstream velos;
    velos << "Tracer" << rank << "_" << tStep << ".vtk";
    rval = mb->write_file( velos.str().c_str(), 0, 0, &euler_set, 1, &tagh, 1 );MB_CHK_ERR( rval );

    return MB_SUCCESS;
}

int main	(	int	argc,
		char **	argv
	)

Definition at line 364 of file diffusion.cpp.

References add_field_value(), ProgOptions::addOpt(), moab::Range::begin(), BRIEF_DESC, moab::ParallelComm::check_all_shared_handles(), compute_tracer_case1(), compute_velocity_case1(), moab::Interface::create_meshset(), IntxUtilsCSLAM::deep_copy_set(), moab::Range::end(), ErrorCode, field_type, moab::Intx2Mesh::FindMaxEdges(), moab::Interface::get_entities_by_dimension(), moab::ParallelComm::get_pcomm(), gtol, input_file, moab::Interface::load_file(), LONG_DESC, mb, MB_CHK_ERR, MB_TAG_CREAT, MB_TAG_DENSE, MB_TYPE_DOUBLE, MESHSET_SET, MPI_COMM_WORLD, numSteps, parallelWrite, moab::ParallelComm::proc_config(), moab::ProcConfig::proc_rank(), radius, rank, moab::Intx2Mesh::set_error_tolerance(), moab::Intx2MeshOnSphere::set_radius_destination_mesh(), moab::Intx2MeshOnSphere::set_radius_source_mesh(), moab::Range::size(), moab::Interface::tag_get_data(), moab::Interface::tag_get_handle(), moab::Interface::tag_iterate(), velocity, and writeFiles.

{
    MPI_Init( &argc, &argv );
    LONG_DESC << "This program simulates a transport problem on a sphere"
                 " according to a benchmark from a Nair & Lauritzen paper.\n"
              << "It starts with a partitioned mesh on a sphere, add a tracer, and steps through.\n"
              << "The flow reverses after half time, and it should return to original "
                 "configuration, if the integration was exact. ";
    ProgOptions opts( LONG_DESC.str(), BRIEF_DESC );

    // read a homme file, partitioned in 16 so far
    std::string fileN          = TestDir + "unittest/mbcslam/fine4.h5m";
    const char* filename_mesh1 = fileN.c_str();

    opts.addOpt< double >( "gtolerance,g", "geometric absolute tolerance (used for point concidence on the sphere)",
                           &gtol );

    std::string input_file;
    opts.addOpt< std::string >( "input_file,i", "input mesh file, partitioned", &input_file );
    std::string extra_read_opts;
    opts.addOpt< std::string >( "extra_read_options,O", "extra read options ", &extra_read_opts );
    // int field_type;
    opts.addOpt< int >( "field_type,f", "field type--  1: quasi-smooth; 2: smooth; 3: slotted cylinders (non-smooth)",
                        &field_type );

    opts.addOpt< int >( "num_steps,n", "number of  steps ", &numSteps );

    // bool reorder = false;
    opts.addOpt< void >( "write_debug_files,w", "write debugging files during simulation ", &writeFiles );

    opts.addOpt< void >( "write_velocity_files,v", "Reorder mesh to group entities by partition", &velocity );

    opts.addOpt< void >( "write_result_in_parallel,p", "write tracer result files", &parallelWrite );

    opts.parseCommandLine( argc, argv );

    if( !input_file.empty() ) filename_mesh1 = input_file.c_str();

    // read in parallel, in the "euler_set", the initial mesh
    std::string optsRead = std::string( "PARALLEL=READ_PART;PARTITION=PARALLEL_PARTITION" ) +
                           std::string( ";PARALLEL_RESOLVE_SHARED_ENTS" ) + extra_read_opts;
    Core moab;
    Interface& mb = moab;
    EntityHandle euler_set;
    ErrorCode rval;
    rval = mb.create_meshset( MESHSET_SET, euler_set );MB_CHK_ERR( rval );

    rval = mb.load_file( filename_mesh1, &euler_set, optsRead.c_str() );MB_CHK_ERR( rval );

    ParallelComm* pcomm = ParallelComm::get_pcomm( &mb, 0 );

    rval = pcomm->check_all_shared_handles();MB_CHK_ERR( rval );

    int rank = pcomm->proc_config().proc_rank();

    if( 0 == rank )
    {
        std::cout << " case 1: use -gtol " << gtol << " -R " << radius << " -input " << filename_mesh1 << " -f "
                  << field_type << " numSteps: " << numSteps << "\n";
        std::cout << " write debug results: " << ( writeFiles ? "yes" : "no" ) << "\n";
        std::cout << " write tracer in parallel: " << ( parallelWrite ? "yes" : "no" ) << "\n";
        std::cout << " output velocity: " << ( velocity ? "yes" : "no" ) << "\n";
    }

    // tagTracer is the value at nodes
    Tag tagTracer = 0;
    std::string tag_name( "Tracer" );
    rval = mb.tag_get_handle( tag_name.c_str(), 1, MB_TYPE_DOUBLE, tagTracer, MB_TAG_DENSE | MB_TAG_CREAT );MB_CHK_ERR( rval );

    // tagElem is the average computed at each element, from nodal values
    Tag tagElem = 0;
    std::string tag_name2( "TracerAverage" );
    rval = mb.tag_get_handle( tag_name2.c_str(), 1, MB_TYPE_DOUBLE, tagElem, MB_TAG_DENSE | MB_TAG_CREAT );MB_CHK_ERR( rval );

    // area of the euler element is fixed, store it; it is used to recompute the averages at each
    // time step
    Tag tagArea = 0;
    std::string tag_name4( "Area" );
    rval = mb.tag_get_handle( tag_name4.c_str(), 1, MB_TYPE_DOUBLE, tagArea, MB_TAG_DENSE | MB_TAG_CREAT );MB_CHK_ERR( rval );

    // add a field value, quasi smooth first
    rval = add_field_value( &mb, euler_set, rank, tagTracer, tagElem, tagArea );MB_CHK_ERR( rval );

    // iniVals are used for 1-norm error computation
    Range redEls;
    rval = mb.get_entities_by_dimension( euler_set, 2, redEls );MB_CHK_ERR( rval );
    std::vector< double > iniVals( redEls.size() );
    rval = mb.tag_get_data( tagElem, redEls, &iniVals[0] );MB_CHK_ERR( rval );

    Tag tagh = 0;
    std::string tag_name3( "Case1" );
    rval = mb.tag_get_handle( tag_name3.c_str(), 3, MB_TYPE_DOUBLE, tagh, MB_TAG_DENSE | MB_TAG_CREAT );MB_CHK_ERR( rval );

    EntityHandle out_set, lagr_set;
    rval = mb.create_meshset( MESHSET_SET, out_set );MB_CHK_ERR( rval );
    rval = mb.create_meshset( MESHSET_SET, lagr_set );MB_CHK_ERR( rval );
    // copy the initial mesh in the lagrangian set
    // initial vertices will be at the same position as euler;

    rval = IntxUtilsCSLAM::deep_copy_set( &mb, euler_set, lagr_set );MB_CHK_ERR( rval );

    Intx2MeshOnSphere worker( &mb );
    worker.set_radius_source_mesh( radius );
    worker.set_radius_destination_mesh( radius );
    worker.set_error_tolerance( gtol );
    worker.set_parallel_comm( pcomm );

    rval = worker.FindMaxEdges( lagr_set, euler_set );MB_CHK_ERR( rval );
    Range local_verts;
    // output also the local_verts
    rval = worker.build_processor_euler_boxes( euler_set, local_verts );MB_CHK_ERR( rval );
    // these stay fixed for one run
    // other things from intersection might need to change, like input blue set (departure set)
    // so we need also a method to clean memory

    for( int i = 1; i < numSteps + 1; i++ )
    {
        // time depends on i; t = i*T/numSteps: ( 0, T/numSteps, 2*T/numSteps, ..., T )
        // this is really just to create some plots; it is not really needed to proceed
        // the compute_tracer_case1 method actually computes the departure point position
        if( velocity )
        {
            rval = compute_velocity_case1( &mb, euler_set, tagh, rank, i );MB_CHK_ERR( rval );
        }

        // this is to actually compute concentrations at time step i, using the
        //  current concentrations
        rval =
            compute_tracer_case1( &mb, worker, euler_set, lagr_set, out_set, tagElem, tagArea, rank, i, local_verts );MB_CHK_ERR( rval );
    }

    // final vals and 1-norm
    Range::iterator iter = redEls.begin();
    double norm1         = 0.;
    int count            = 0;
    void* data;
    int j = 0;  // index in iniVals
    while( iter != redEls.end() )
    {
        rval = mb.tag_iterate( tagElem, iter, redEls.end(), count, data );MB_CHK_ERR( rval );
        double* ptrTracer = (double*)data;

        rval = mb.tag_iterate( tagArea, iter, redEls.end(), count, data );MB_CHK_ERR( rval );
        double* ptrArea = (double*)data;
        for( int i = 0; i < count; i++, ++iter, ptrTracer++, ptrArea++, j++ )
        {
            // double area = *ptrArea;
            norm1 += fabs( *ptrTracer - iniVals[j] ) * ( *ptrArea );
        }
    }

    double total_norm1 = 0;
    int mpi_err        = MPI_Reduce( &norm1, &total_norm1, 1, MPI_DOUBLE, MPI_SUM, 0, MPI_COMM_WORLD );
    if( MPI_SUCCESS != mpi_err ) return 1;
    if( 0 == rank ) std::cout << " numSteps:" << numSteps << " 1-norm:" << total_norm1 << "\n";
    MPI_Finalize();
    return 0;
}