Add example elliptic1DLeftNeumannRightNeumann.cpp

doc/sphinx/source/examples/Elliptic/1D/Elliptic1D-Left-Neumann-Right-Neumann.md

@@ -47,6 +47,7 @@ $$
 
 This example is implemented in:
 - [MATLAB/ OCTAVE](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab_octave/elliptic1DLeftNeumannRightNeumann.m)
+- [C++](https://github.com/csrc-sdsu/mole/blob/main/examples/cpp/elliptic1DLeftNeumannRightNeumann.cpp)
 
 Additional MATLAB/ OCTAVE variants of this example with different boundary conditions:
 - [Homogeneous Dirichlet](https://github.com/csrc-sdsu/mole/blob/main/examples/matlab_octave/elliptic1DHomogeneousDirichlet.m)

examples/cpp/elliptic1DLeftNeumannRightNeumann.cpp

@@ -0,0 +1,93 @@
+/**
+ * @file elliptic1DLeftNeumannRightNeumann.cpp
+ * @brief Solves the 1D linear equation -u'' = x - 1/2
+ *
+ * ## Spatial Domains:
+ * - The spatial domain is [0, 1]
+ * - Interior points are spaced by dx = (b - a) / m.
+ *
+ * ## Boundary Conditions:
+ *      u'(0) = 0, u'(1) = 0
+ *
+ * The solution is computed numerically, and the result is compared with the exact solution:
+ *      u_exact(x) = x^2/4 - x^3/6
+ *
+ * The results are saved to a file "plot.gnu" and visualized using GNUplot.
+*/
+
+#include "mole.h"
+#include <iostream>
+
+int main() {
+
+    const int k = 2;
+    const int m = 10;
+    const Real dx = 1.0 / m;
+    const Real a = 0;   // left boundary
+    const Real b = 1;   // right boundary
+
+    // Mimetic operators
+    Laplacian L(k, m, dx);
+    const Real d = 0; // Dirichlet coeff
+    const Real n = 1; // Neumann coeff
+    RobinBC BC(k, m, dx, d, n);
+    L += BC;
+
+    // 1D grid
+    arma::vec grid(m + 2);
+    grid(0) = a;
+    grid(1) = grid(0) + dx / 2.0;
+    for (int i = 2; i <= m; i++) {
+        grid(i) = grid(i - 1) + dx;
+    }
+    grid(m + 1) = b;
+
+    // RHS
+    arma::vec rhs(m+2); rhs.zeros();
+    for (int i=1; i<=m; ++i) {
+        rhs(i) = -(grid(i) - 0.5);
+    }
+    rhs(0) = 0.0;
+    rhs(m+1) = 0.0;
+
+    // Solve the system
+    #ifdef EIGEN
+        // Use eigen if available
+        arma::vec sol = Utils:: spsolve_eigen(L, rhs);
+    #else
+        arma::vec sol = spsolve(L, rhs);
+    #endif
+
+    // shift
+    const Real val = sol(0);
+    for (int j = 0; j < m+2; ++j) {
+        sol(j) -= val;
+    }
+
+    // Create a GNUplot script file
+    std::ofstream plot_script("plot.gnu");
+    if (!plot_script) {
+        std::cerr << "Error: Failed to create GNUplot script.\n";
+        return 1;
+    }
+    plot_script << "set title \"-u'' = x - 1/2, u'(0) = 0, u'(1) = 0\"\n";
+    plot_script << "set xlabel 't'\n";
+    plot_script << "set ylabel 'y'\n";
+    plot_script << "plot '-' using 1:2 with lines title \"Estimated Solution\", "
+                << "x**2 / 4 - x**3 / 6 with lines title \"Exact Solution\"\n";
+
+    for (int i = 0; i <= m + 1; ++i) {
+        plot_script << grid(i) << " " << sol(i) << "\n";
+    }
+    plot_script.close();
+
+    // Execute GNUplot using the script
+    if (system("gnuplot -persist plot.gnu") != 0) {
+        std::cerr << "Error: Failed to execute GNUplot.\n";
+        return 1;
+    }
+
+    cout << sol;
+
+    return 0;
+}