MeanField/tests/mapping/compactification/kelvin.cpp

#include <array>
#include <catch2/catch_test_macros.hpp>
#include <catch2/matchers/catch_matchers_floating_point.hpp>
#include <cmath>
#include <limits>
#include <mfem.hpp>
#include <stdexcept>

import mean_field;
import test_helpers;

using namespace mean_field;
using Catch::Matchers::WithinAbs;

namespace {
    constexpr int dimension = 3;

    mfem::Vector make_vector(
        const double x,
        const double y,
        const double z
    ) {
        mfem::Vector vector(dimension);
        vector(0) = x;
        vector(1) = y;
        vector(2) = z;
        return vector;
    }

    mfem::DenseMatrix make_identity() {
        mfem::DenseMatrix matrix(dimension);
        matrix = 0.0;
        for (int i = 0; i < dimension; ++i)
            matrix(i, i) = 1.0;
        return matrix;
    }

    void check_vector(
        const mfem::Vector &actual,
        const mfem::Vector &expected,
        const double tolerance
    ) {
        REQUIRE(actual.Size() == expected.Size());
        for (int i = 0; i < actual.Size(); ++i)
            CHECK_THAT(actual(i), WithinAbs(expected(i), tolerance));
    }

    void check_matrix(
        const mfem::DenseMatrix &actual,
        const mfem::DenseMatrix &expected,
        const double tolerance
    ) {
        REQUIRE(actual.Height() == expected.Height());
        REQUIRE(actual.Width() == expected.Width());

        for (int i = 0; i < actual.Height(); ++i) {
            for (int j = 0; j < actual.Width(); ++j)
                CHECK_THAT(actual(i, j), WithinAbs(expected(i, j), tolerance));
        }
    }

    struct AnalyticFactors {
        double computational_radius;
        double scale;
        double scale_derivative;
    };

    AnalyticFactors compute_analytic_factors(
        const double r_star,
        const double r_inf,
        const double coordinate
    ) {
        const double radial_extent        = r_inf - r_star;
        const double computational_radius = r_star + coordinate * radial_extent;
        const double scale                = r_star / (computational_radius * (1.0 - coordinate));
        const double scale_derivative     = scale * (1.0 / (1.0 - coordinate) - radial_extent / computational_radius);
        return {.computational_radius = computational_radius, .scale = scale, .scale_derivative = scale_derivative};
    }

    mapping::MappingStatus evaluate_affine_map(
        const mapping::compactification::ExteriorDomainMap &exterior_map,
        const mfem::Vector &reference_position,
        const mfem::DenseMatrix &affine_jacobian,
        const mfem::Vector &offset,
        const double compactification_coordinate,
        const mfem::Vector &compactification_coordinate_gradient,
        mapping::compactification::ExteriorMapResult &result
    ) {
        mfem::Vector displaced_position(dimension);
        affine_jacobian.Mult(reference_position, displaced_position);
        displaced_position += offset;

        const mapping::compactification::ExteriorMapInput input{
            .reference_position                   = reference_position,
            .displaced_position                   = displaced_position,
            .displacement_jacobian                = affine_jacobian,
            .compactification_coordinate          = compactification_coordinate,
            .compactification_coordinate_gradient = compactification_coordinate_gradient
        };

        return exterior_map.Evaluate(input, result);
    }
} // namespace

TEST_CASE(
    "Kelvin Compactification Validates Its Configuration",
    tags::unit &tags::mapping &tags::kelvin
) {
    CHECK_NOTHROW(mapping::compactification::KelvinCompactification({.r_star_ref = 1.0, .r_inf_ref = 4.0}));
    CHECK_THROWS_AS(
        mapping::compactification::KelvinCompactification({.r_star_ref = 0.0, .r_inf_ref = 4.0}), std::invalid_argument
    );
    CHECK_THROWS_AS(
        mapping::compactification::KelvinCompactification({.r_star_ref = -1.0, .r_inf_ref = 4.0}), std::invalid_argument
    );
    CHECK_THROWS_AS(
        mapping::compactification::KelvinCompactification({.r_star_ref = 2.0, .r_inf_ref = 2.0}), std::invalid_argument
    );
    CHECK_THROWS_AS(
        mapping::compactification::KelvinCompactification({.r_star_ref = 3.0, .r_inf_ref = 2.0}), std::invalid_argument
    );
    CHECK_THROWS_AS(
        mapping::compactification::KelvinCompactification(
            {.r_star_ref = 1.0, .r_inf_ref = std::numeric_limits<double>::infinity()}
        ),
        std::invalid_argument
    );
    CHECK_THROWS_AS(
        mapping::compactification::KelvinCompactification(
            {.r_star_ref = 1.0, .r_inf_ref = 4.0, .coordinate_tolerance = -1.0e-12}
        ),
        std::invalid_argument
    );
    CHECK_THROWS_AS(
        mapping::compactification::KelvinCompactification(
            {.r_star_ref = 1.0, .r_inf_ref = 4.0, .coordinate_tolerance = 1.0}
        ),
        std::invalid_argument
    );
    CHECK_THROWS_AS(
        mapping::compactification::KelvinCompactification(
            {.r_star_ref = 1.0, .r_inf_ref = 4.0, .coordinate_tolerance = std::numeric_limits<double>::quiet_NaN()}
        ),
        std::invalid_argument
    );
}

TEST_CASE(
    "Kelvin Compactification Reports Its Configuration",
    tags::unit &tags::mapping &tags::kelvin
) {
    constexpr double coordinate_tolerance = 3.0e-11;
    mapping::compactification::KelvinCompactification compactification(
        {.r_star_ref = 1.25, .r_inf_ref = 5.5, .coordinate_tolerance = coordinate_tolerance}
    );

    CHECK(compactification.GetName() == "KelvinCompactification");
    CHECK_THAT(compactification.GetReferenceStellarRadius(), WithinAbs(1.25, 0.0));
    CHECK_THAT(compactification.GetReferenceInfinityRadius(), WithinAbs(5.5, 0.0));
    CHECK_THAT(compactification.GetCoordinateTolerance(), WithinAbs(coordinate_tolerance, 0.0));
}

TEST_CASE(
    "Kelvin Compactification Uses The Exterior Coordinate Rather Than "
    "Euclidean Radius",
    tags::unit &tags::mapping &tags::kelvin
) {
    constexpr double tolerance  = 0.0;
    constexpr double coordinate = 0.37;

    mapping::compactification::KelvinCompactification compactification({.r_star_ref = 1.0, .r_inf_ref = 4.0});
    const mfem::DenseMatrix displacement_jacobian = make_identity();
    const mfem::Vector displaced_position         = make_vector(1.4, -0.2, 0.3);
    const mfem::Vector coordinate_gradient        = make_vector(0.2, -0.1, 0.05);
    const mfem::Vector reference_a                = make_vector(0.2, 0.1, -0.1);
    const mfem::Vector reference_b                = make_vector(12.0, -7.0, 4.0);

    const mapping::compactification::ExteriorMapInput input_a{
        reference_a, displaced_position, displacement_jacobian, coordinate, coordinate_gradient
    };
    const mapping::compactification::ExteriorMapInput input_b{
        reference_b, displaced_position, displacement_jacobian, coordinate, coordinate_gradient
    };

    mapping::compactification::ExteriorMapResult result_a;
    mapping::compactification::ExteriorMapResult result_b;

    REQUIRE(compactification.Evaluate(input_a, result_a) == mapping::MappingStatus::valid);
    REQUIRE(compactification.Evaluate(input_b, result_b) == mapping::MappingStatus::valid);

    check_vector(result_a.physical_position, result_b.physical_position, tolerance);
    check_matrix(result_a.mapping_jacobian, result_b.mapping_jacobian, tolerance);
}

TEST_CASE(
    "Kelvin Compactification Matches Its Analytic Radial Map",
    tags::unit &tags::mapping &tags::kelvin
) {
    constexpr double r_star        = 1.0;
    constexpr double r_inf         = 4.0;
    constexpr double radial_extent = r_inf - r_star;
    constexpr double tolerance     = 2.0e-12;

    mapping::compactification::KelvinCompactification compactification({.r_star_ref = r_star, .r_inf_ref = r_inf});
    const mfem::DenseMatrix identity       = make_identity();
    const mfem::Vector coordinate_gradient = make_vector(1.0 / radial_extent, 0.0, 0.0);

    for (const double computational_radius : std::array{1.0, 1.25, 2.0, 3.0, 3.75}) {
        CAPTURE(computational_radius);

        const double coordinate               = (computational_radius - r_star) / radial_extent;
        const mfem::Vector reference_position = make_vector(computational_radius, 0.0, 0.0);
        const mfem::Vector displaced_position(reference_position);
        const mapping::compactification::ExteriorMapInput input{
            reference_position, displaced_position, identity, coordinate, coordinate_gradient
        };
        mapping::compactification::ExteriorMapResult result;

        REQUIRE(compactification.Evaluate(input, result) == mapping::MappingStatus::valid);

        const double expected_radius            = r_star * radial_extent / (r_inf - computational_radius);
        const double expected_radial_derivative = r_star * radial_extent / std::pow(r_inf - computational_radius, 2.0);
        const double expected_tangential_scale  = expected_radius / computational_radius;

        const mfem::Vector expected_position    = make_vector(expected_radius, 0.0, 0.0);
        mfem::DenseMatrix expected_jacobian(dimension);
        expected_jacobian       = 0.0;
        expected_jacobian(0, 0) = expected_radial_derivative;
        expected_jacobian(1, 1) = expected_tangential_scale;
        expected_jacobian(2, 2) = expected_tangential_scale;

        check_vector(result.physical_position, expected_position, tolerance);
        check_matrix(result.mapping_jacobian, expected_jacobian, tolerance);
        CHECK(result.mapping_jacobian.Det() > 0.0);
    }
}

TEST_CASE(
    "Kelvin Compactification Matches Its Full Cartesian Formula",
    tags::unit &tags::mapping &tags::kelvin
) {
    constexpr double r_star     = 1.0;
    constexpr double r_inf      = 4.0;
    constexpr double coordinate = 0.42;
    constexpr double tolerance  = 1.0e-12;

    mapping::compactification::KelvinCompactification compactification({.r_star_ref = r_star, .r_inf_ref = r_inf});
    const mfem::Vector reference_position  = make_vector(0.8, 0.4, -0.2);
    const mfem::Vector displaced_position  = make_vector(1.1, 0.5, -0.1);
    const mfem::Vector coordinate_gradient = make_vector(0.20, -0.10, 0.05);

    mfem::DenseMatrix displacement_jacobian(dimension);
    displacement_jacobian(0, 0) = 1.10;
    displacement_jacobian(0, 1) = 0.05;
    displacement_jacobian(0, 2) = 0.00;
    displacement_jacobian(1, 0) = -0.02;
    displacement_jacobian(1, 1) = 0.95;
    displacement_jacobian(1, 2) = 0.03;
    displacement_jacobian(2, 0) = 0.01;
    displacement_jacobian(2, 1) = -0.04;
    displacement_jacobian(2, 2) = 1.05;

    const mapping::compactification::ExteriorMapInput input{
        reference_position, displaced_position, displacement_jacobian, coordinate, coordinate_gradient
    };
    mapping::compactification::ExteriorMapResult result;

    REQUIRE(compactification.Evaluate(input, result) == mapping::MappingStatus::valid);

    const AnalyticFactors factors = compute_analytic_factors(r_star, r_inf, coordinate);
    mfem::Vector expected_position(displaced_position);
    expected_position *= factors.scale;

    mfem::DenseMatrix expected_jacobian(displacement_jacobian);
    expected_jacobian *= factors.scale;

    for (int i = 0; i < dimension; ++i) {
        for (int j = 0; j < dimension; ++j)
            expected_jacobian(i, j) += displaced_position(i) * factors.scale_derivative * coordinate_gradient(j);
    }

    check_vector(result.physical_position, expected_position, tolerance);
    check_matrix(result.mapping_jacobian, expected_jacobian, tolerance);
}

TEST_CASE(
    "Kelvin Compactification Jacobian Matches Coordinate Finite Differences",
    tags::unit &tags::mapping &tags::kelvin
) {
    constexpr double base_coordinate = 0.38;
    constexpr double difference_step = 1.0e-6;
    constexpr double tolerance       = 3.0e-9;

    mapping::compactification::KelvinCompactification compactification({.r_star_ref = 1.0, .r_inf_ref = 4.0});

    mfem::DenseMatrix affine_jacobian(dimension);
    affine_jacobian(0, 0)                  = 1.05;
    affine_jacobian(0, 1)                  = 0.02;
    affine_jacobian(0, 2)                  = 0.00;
    affine_jacobian(1, 0)                  = -0.01;
    affine_jacobian(1, 1)                  = 0.98;
    affine_jacobian(1, 2)                  = 0.03;
    affine_jacobian(2, 0)                  = 0.02;
    affine_jacobian(2, 1)                  = 0.00;
    affine_jacobian(2, 2)                  = 1.02;

    const mfem::Vector offset              = make_vector(0.04, -0.03, 0.02);
    const mfem::Vector reference_position  = make_vector(1.4, 0.3, -0.2);
    const mfem::Vector coordinate_gradient = make_vector(0.11, -0.07, 0.05);

    mapping::compactification::ExteriorMapResult base_result;
    REQUIRE(
        evaluate_affine_map(
            compactification, reference_position, affine_jacobian, offset, base_coordinate, coordinate_gradient,
            base_result
        ) == mapping::MappingStatus::valid
    );

    for (int coordinate = 0; coordinate < dimension; ++coordinate) {
        mfem::Vector reference_plus(reference_position);
        mfem::Vector reference_minus(reference_position);
        reference_plus(coordinate) += difference_step;
        reference_minus(coordinate) -= difference_step;

        const double compactification_plus  = base_coordinate + difference_step * coordinate_gradient(coordinate);
        const double compactification_minus = base_coordinate - difference_step * coordinate_gradient(coordinate);

        mapping::compactification::ExteriorMapResult result_plus;
        mapping::compactification::ExteriorMapResult result_minus;

        REQUIRE(
            evaluate_affine_map(
                compactification, reference_plus, affine_jacobian, offset, compactification_plus, coordinate_gradient,
                result_plus
            ) == mapping::MappingStatus::valid
        );
        REQUIRE(
            evaluate_affine_map(
                compactification, reference_minus, affine_jacobian, offset, compactification_minus, coordinate_gradient,
                result_minus
            ) == mapping::MappingStatus::valid
        );

        for (int component = 0; component < dimension; ++component) {
            const double finite_difference =
                (result_plus.physical_position(component) - result_minus.physical_position(component)) /
                (2.0 * difference_step);
            CHECK_THAT(finite_difference, WithinAbs(base_result.mapping_jacobian(component, coordinate), tolerance));
        }
    }
}

TEST_CASE(
    "Kelvin Compactification Variation Matches State Finite Differences",
    tags::unit &tags::mapping &tags::kelvin
) {
    constexpr double coordinate      = 0.46;
    constexpr double difference_step = 1.0e-6;
    constexpr double tolerance       = 2.0e-10;

    mapping::compactification::KelvinCompactification compactification({.r_star_ref = 1.0, .r_inf_ref = 4.0});
    const mfem::Vector reference_position   = make_vector(1.3, -0.2, 0.4);
    const mfem::Vector displaced_position   = make_vector(1.4, -0.1, 0.35);
    const mfem::Vector coordinate_gradient  = make_vector(0.12, -0.04, 0.08);
    const mfem::Vector position_direction   = make_vector(0.03, -0.05, 0.02);

    mfem::DenseMatrix displacement_jacobian = make_identity();
    displacement_jacobian(0, 1)             = 0.04;
    displacement_jacobian(1, 2)             = -0.03;
    displacement_jacobian(2, 0)             = 0.02;

    mfem::DenseMatrix jacobian_direction(dimension);
    jacobian_direction(0, 0) = 0.02;
    jacobian_direction(0, 1) = -0.01;
    jacobian_direction(0, 2) = 0.03;
    jacobian_direction(1, 0) = 0.01;
    jacobian_direction(1, 1) = -0.02;
    jacobian_direction(1, 2) = 0.00;
    jacobian_direction(2, 0) = -0.01;
    jacobian_direction(2, 1) = 0.02;
    jacobian_direction(2, 2) = 0.01;

    const mapping::compactification::ExteriorMapInput input{
        reference_position, displaced_position, displacement_jacobian, coordinate, coordinate_gradient
    };
    const mapping::compactification::ExteriorMapDirection direction{position_direction, jacobian_direction};

    mapping::compactification::ExteriorMapResult base_result;
    mapping::compactification::ExteriorMapVariation variation;

    REQUIRE(compactification.Evaluate(input, base_result) == mapping::MappingStatus::valid);
    REQUIRE(
        compactification.EvaluateVariation(input, base_result, direction, variation) == mapping::MappingStatus::valid
    );

    mfem::Vector displaced_plus(displaced_position);
    mfem::Vector displaced_minus(displaced_position);
    displaced_plus.Add(difference_step, position_direction);
    displaced_minus.Add(-difference_step, position_direction);

    mfem::DenseMatrix jacobian_plus(displacement_jacobian);
    mfem::DenseMatrix jacobian_minus(displacement_jacobian);
    jacobian_plus.Add(difference_step, jacobian_direction);
    jacobian_minus.Add(-difference_step, jacobian_direction);

    const mapping::compactification::ExteriorMapInput input_plus{
        reference_position, displaced_plus, jacobian_plus, coordinate, coordinate_gradient
    };
    const mapping::compactification::ExteriorMapInput input_minus{
        reference_position, displaced_minus, jacobian_minus, coordinate, coordinate_gradient
    };

    mapping::compactification::ExteriorMapResult result_plus;
    mapping::compactification::ExteriorMapResult result_minus;

    REQUIRE(compactification.Evaluate(input_plus, result_plus) == mapping::MappingStatus::valid);
    REQUIRE(compactification.Evaluate(input_minus, result_minus) == mapping::MappingStatus::valid);

    for (int i = 0; i < dimension; ++i) {
        const double position_finite_difference =
            (result_plus.physical_position(i) - result_minus.physical_position(i)) / (2.0 * difference_step);
        CHECK_THAT(position_finite_difference, WithinAbs(variation.physical_position_variation(i), tolerance));

        for (int j = 0; j < dimension; ++j) {
            const double jacobian_finite_difference =
                (result_plus.mapping_jacobian(i, j) - result_minus.mapping_jacobian(i, j)) / (2.0 * difference_step);
            CHECK_THAT(jacobian_finite_difference, WithinAbs(variation.mapping_jacobian_variation(i, j), tolerance));
        }
    }
}

TEST_CASE(
    "Kelvin Compactification Preserves Rotational Covariance",
    tags::unit &tags::mapping &tags::kelvin
) {
    constexpr double coordinate = 0.31;
    constexpr double tolerance  = 1.0e-12;

    mapping::compactification::KelvinCompactification compactification({.r_star_ref = 1.0, .r_inf_ref = 4.0});
    const mfem::Vector reference_position   = make_vector(1.3, 0.4, -0.2);
    const mfem::Vector displaced_position   = make_vector(1.4, 0.2, -0.1);
    const mfem::Vector coordinate_gradient  = make_vector(0.16, -0.08, 0.03);

    mfem::DenseMatrix displacement_jacobian = make_identity();
    displacement_jacobian(0, 1)             = 0.05;
    displacement_jacobian(1, 0)             = -0.02;
    displacement_jacobian(2, 1)             = 0.03;

    const mapping::compactification::ExteriorMapInput input{
        reference_position, displaced_position, displacement_jacobian, coordinate, coordinate_gradient
    };
    mapping::compactification::ExteriorMapResult result;
    REQUIRE(compactification.Evaluate(input, result) == mapping::MappingStatus::valid);

    mfem::DenseMatrix rotation(dimension);
    rotation       = 0.0;
    rotation(0, 1) = -1.0;
    rotation(1, 0) = 1.0;
    rotation(2, 2) = 1.0;

    mfem::Vector rotated_reference(dimension);
    mfem::Vector rotated_displaced(dimension);
    mfem::Vector rotated_coordinate_gradient(dimension);
    rotation.Mult(reference_position, rotated_reference);
    rotation.Mult(displaced_position, rotated_displaced);
    rotation.Mult(coordinate_gradient, rotated_coordinate_gradient);

    mfem::DenseMatrix temporary(dimension);
    mfem::DenseMatrix rotated_displacement_jacobian(dimension);
    mfem::Mult(rotation, displacement_jacobian, temporary);
    mfem::MultABt(temporary, rotation, rotated_displacement_jacobian);

    const mapping::compactification::ExteriorMapInput rotated_input{
        rotated_reference, rotated_displaced, rotated_displacement_jacobian, coordinate, rotated_coordinate_gradient
    };
    mapping::compactification::ExteriorMapResult rotated_result;
    REQUIRE(compactification.Evaluate(rotated_input, rotated_result) == mapping::MappingStatus::valid);

    mfem::Vector expected_position(dimension);
    rotation.Mult(result.physical_position, expected_position);

    mfem::DenseMatrix expected_jacobian(dimension);
    mfem::Mult(rotation, result.mapping_jacobian, temporary);
    mfem::MultABt(temporary, rotation, expected_jacobian);

    check_vector(rotated_result.physical_position, expected_position, tolerance);
    check_matrix(rotated_result.mapping_jacobian, expected_jacobian, tolerance);
}

TEST_CASE(
    "Kelvin Compactification Is Continuous At The Mesh Defined Stellar Surface",
    tags::unit &tags::mapping &tags::kelvin
) {
    constexpr double tolerance = 1.0e-14;

    mapping::compactification::KelvinCompactification compactification({.r_star_ref = 1.0, .r_inf_ref = 4.0});
    const mfem::Vector reference_position  = make_vector(-0.5260553366425769, 0.5260553366425769, -0.6553163792879153);
    const mfem::Vector displaced_position  = make_vector(-0.55, 0.51, -0.63);
    const mfem::Vector coordinate_gradient = make_vector(-0.18, 0.18, -0.22);
    const mfem::DenseMatrix displacement_jacobian = make_identity();

    REQUIRE(reference_position.Norml2() < 1.0);

    const mapping::compactification::ExteriorMapInput input{
        reference_position, displaced_position, displacement_jacobian, 0.0, coordinate_gradient
    };
    mapping::compactification::ExteriorMapResult result;

    REQUIRE(compactification.Evaluate(input, result) == mapping::MappingStatus::valid);
    check_vector(result.physical_position, displaced_position, tolerance);
    CHECK(result.mapping_jacobian.Det() > 0.0);
}

TEST_CASE(
    "Kelvin Compactification Has Correct Infinity And Coordinate Bound "
    "Behavior",
    tags::unit &tags::mapping &tags::kelvin
) {
    constexpr double r_star               = 1.0;
    constexpr double r_inf                = 4.0;
    constexpr double radial_extent        = r_inf - r_star;
    constexpr double coordinate_tolerance = 1.0e-12;
    constexpr double tolerance            = 2.0e-11;

    mapping::compactification::KelvinCompactification compactification(
        {.r_star_ref = r_star, .r_inf_ref = r_inf, .coordinate_tolerance = coordinate_tolerance}
    );
    const mfem::DenseMatrix identity       = make_identity();
    const mfem::Vector coordinate_gradient = make_vector(1.0 / radial_extent, 0.0, 0.0);

    for (const double coordinate : std::array{0.0, 0.25, 0.75, 0.95, 0.99, 0.999}) {
        CAPTURE(coordinate);

        const double computational_radius     = r_star + coordinate * radial_extent;
        const mfem::Vector reference_position = make_vector(computational_radius, 0.0, 0.0);
        const mapping::compactification::ExteriorMapInput input{
            reference_position, reference_position, identity, coordinate, coordinate_gradient
        };
        mapping::compactification::ExteriorMapResult result;

        REQUIRE(compactification.Evaluate(input, result) == mapping::MappingStatus::valid);
        CHECK_THAT(result.physical_position.Norml2() * (1.0 - coordinate), WithinAbs(r_star, tolerance));
    }

    const mfem::Vector reference_position = make_vector(r_inf, 0.0, 0.0);
    mapping::compactification::ExteriorMapResult result;

    CHECK(
        compactification.Evaluate(
            {reference_position, reference_position, identity, 1.0, coordinate_gradient}, result
        ) == mapping::MappingStatus::at_compactified_infinity
    );
    CHECK(
        compactification.Evaluate(
            {reference_position, reference_position, identity, 1.0 - 0.5 * coordinate_tolerance, coordinate_gradient},
            result
        ) == mapping::MappingStatus::at_compactified_infinity
    );
    CHECK(
        compactification.Evaluate(
            {reference_position, reference_position, identity, 1.0 + 0.5 * coordinate_tolerance, coordinate_gradient},
            result
        ) == mapping::MappingStatus::at_compactified_infinity
    );
    CHECK(
        compactification.Evaluate(
            {reference_position, reference_position, identity, 1.0 + 2.0 * coordinate_tolerance, coordinate_gradient},
            result
        ) == mapping::MappingStatus::outside_reference_domain
    );
    CHECK(
        compactification.Evaluate(
            {reference_position, reference_position, identity, -2.0 * coordinate_tolerance, coordinate_gradient}, result
        ) == mapping::MappingStatus::outside_reference_domain
    );

    const mfem::Vector surface_position = make_vector(0.97, 0.0, 0.0);
    CHECK(
        compactification.Evaluate(
            {surface_position, surface_position, identity, -0.5 * coordinate_tolerance, coordinate_gradient}, result
        ) == mapping::MappingStatus::valid
    );
    check_vector(result.physical_position, surface_position, tolerance);
}

TEST_CASE(
    "Kelvin Compactification Rejects Invalid Inputs And Inverted Maps",
    tags::unit &tags::mapping &tags::kelvin
) {
    mapping::compactification::KelvinCompactification compactification({.r_star_ref = 1.0, .r_inf_ref = 4.0});

    const mfem::Vector reference_position = make_vector(2.0, 0.0, 0.0);
    const mfem::Vector displaced_position(reference_position);
    const mfem::Vector coordinate_gradient = make_vector(1.0 / 3.0, 0.0, 0.0);
    const mfem::Vector zero_gradient       = make_vector(0.0, 0.0, 0.0);
    const mfem::DenseMatrix identity       = make_identity();
    mapping::compactification::ExteriorMapResult result;

    mfem::Vector wrong_dimension(2);
    wrong_dimension = 1.0;
    CHECK(
        compactification.Evaluate(
            {wrong_dimension, displaced_position, identity, 1.0 / 3.0, coordinate_gradient}, result
        ) == mapping::MappingStatus::invalid_dimension
    );
    CHECK(
        compactification.Evaluate(
            {reference_position, displaced_position, identity, 1.0 / 3.0, wrong_dimension}, result
        ) == mapping::MappingStatus::invalid_dimension
    );

    mfem::Vector non_finite_position(reference_position);
    non_finite_position(1) = std::numeric_limits<double>::quiet_NaN();
    CHECK(
        compactification.Evaluate(
            {non_finite_position, displaced_position, identity, 1.0 / 3.0, coordinate_gradient}, result
        ) == mapping::MappingStatus::non_finite_input
    );

    mfem::Vector non_finite_gradient(coordinate_gradient);
    non_finite_gradient(2) = std::numeric_limits<double>::infinity();
    CHECK(
        compactification.Evaluate(
            {reference_position, displaced_position, identity, 1.0 / 3.0, non_finite_gradient}, result
        ) == mapping::MappingStatus::non_finite_input
    );
    CHECK(
        compactification.Evaluate(
            {reference_position, displaced_position, identity, std::numeric_limits<double>::quiet_NaN(),
             coordinate_gradient},
            result
        ) == mapping::MappingStatus::non_finite_input
    );

    mfem::DenseMatrix singular_displacement_jacobian(dimension);
    singular_displacement_jacobian = 0.0;
    CHECK(
        compactification.Evaluate(
            {reference_position, displaced_position, singular_displacement_jacobian, 1.0 / 3.0, zero_gradient}, result
        ) == mapping::MappingStatus::non_positive_determinant
    );

    mfem::DenseMatrix inverted_displacement_jacobian = make_identity();
    inverted_displacement_jacobian(0, 0)             = -1.0;
    CHECK(
        compactification.Evaluate(
            {reference_position, displaced_position, inverted_displacement_jacobian, 1.0 / 3.0, zero_gradient}, result
        ) == mapping::MappingStatus::non_positive_determinant
    );
}

TEST_CASE(
    "Kelvin Compactification Variation Rejects Invalid Inputs",
    tags::unit &tags::mapping &tags::kelvin
) {
    mapping::compactification::KelvinCompactification compactification({.r_star_ref = 1.0, .r_inf_ref = 4.0});
    const mfem::Vector reference_position = make_vector(2.0, 0.0, 0.0);
    const mfem::Vector displaced_position(reference_position);
    const mfem::Vector coordinate_gradient = make_vector(1.0 / 3.0, 0.0, 0.0);
    const mfem::DenseMatrix identity       = make_identity();
    const mapping::compactification::ExteriorMapInput input{
        reference_position, displaced_position, identity, 1.0 / 3.0, coordinate_gradient
    };

    mapping::compactification::ExteriorMapResult result;
    REQUIRE(compactification.Evaluate(input, result) == mapping::MappingStatus::valid);

    const mfem::Vector valid_position_direction      = make_vector(0.01, -0.02, 0.03);
    const mfem::DenseMatrix valid_jacobian_direction = make_identity();
    mapping::compactification::ExteriorMapVariation variation;

    mfem::Vector wrong_dimension(2);
    wrong_dimension = 0.0;
    CHECK(
        compactification.EvaluateVariation(input, result, {wrong_dimension, valid_jacobian_direction}, variation) ==
        mapping::MappingStatus::invalid_dimension
    );

    mfem::DenseMatrix wrong_jacobian_dimension(2);
    wrong_jacobian_dimension = 0.0;
    CHECK(
        compactification.EvaluateVariation(
            input, result, {valid_position_direction, wrong_jacobian_dimension}, variation
        ) == mapping::MappingStatus::invalid_dimension
    );

    mfem::Vector non_finite_direction(valid_position_direction);
    non_finite_direction(0) = std::numeric_limits<double>::quiet_NaN();
    CHECK(
        compactification.EvaluateVariation(
            input, result, {non_finite_direction, valid_jacobian_direction}, variation
        ) == mapping::MappingStatus::non_finite_input
    );
}

TEST_CASE(
    "Kelvin Compactification Evaluations Are Independent",
    tags::unit &tags::mapping &tags::kelvin
) {
    constexpr double tolerance = 0.0;

    mapping::compactification::KelvinCompactification compactification({.r_star_ref = 1.0, .r_inf_ref = 4.0});
    const mfem::DenseMatrix identity = make_identity();
    const mfem::Vector gradient_a    = make_vector(0.12, 0.03, -0.02);
    const mfem::Vector gradient_b    = make_vector(-0.04, 0.15, 0.01);
    const mfem::Vector reference_a   = make_vector(1.5, 0.2, 0.1);
    const mfem::Vector reference_b   = make_vector(2.5, -0.3, 0.4);

    const mapping::compactification::ExteriorMapInput input_a{reference_a, reference_a, identity, 0.25, gradient_a};
    const mapping::compactification::ExteriorMapInput input_b{reference_b, reference_b, identity, 0.70, gradient_b};

    mapping::compactification::ExteriorMapResult first_a;
    mapping::compactification::ExteriorMapResult result_b;
    mapping::compactification::ExteriorMapResult second_a;

    REQUIRE(compactification.Evaluate(input_a, first_a) == mapping::MappingStatus::valid);
    REQUIRE(compactification.Evaluate(input_b, result_b) == mapping::MappingStatus::valid);
    REQUIRE(compactification.Evaluate(input_a, second_a) == mapping::MappingStatus::valid);

    check_vector(first_a.physical_position, second_a.physical_position, tolerance);
    check_matrix(first_a.mapping_jacobian, second_a.mapping_jacobian, tolerance);
}