blowing up error when adding second order AB method

src/Numerics/OdeSolvers/AdamsBashforth.cs

@@ -27,15 +27,11 @@
 // OTHER DEALINGS IN THE SOFTWARE.
 // </copyright>
 using System;
-using System.Collections.Generic;
-using System.Linq;
-using System.Text;
 
 namespace MathNet.Numerics.OdeSolvers
 {
     public static class AdamsBashforth
     {
-
         /// <summary>
         /// First Order AB method(same as Forward Euler)
         /// </summary>
@@ -59,10 +55,32 @@ namespace MathNet.Numerics.OdeSolvers
             }
             return y;
         }
-
-        public static double[] SecondOrder()
+        /// <summary>
+        /// Second Order AB Method(Require two initial guesses)
+        /// </summary>
+        /// <param name="y0">Initial value 1</param>
+        /// <param name="y1">Initial value 2</param>
+        /// <param name="start">Start Time</param>
+        /// <param name="end">End Time</param>
+        /// <param name="N">Number of subintervals</param>
+        /// <param name="f">ode model</param>
+        /// <returns></returns>
+        public static double[] SecondOrder(double y0, double y1, double start, double end, int N, Func<double, double, double> f)
         {
-            return null;
+            double dt = (end - start) / (N - 1);
+            double t = start;
+            double[] y = new double[N];
+            y[0] = y0;
+            y[1] = y1;
+            for (int i = 2; i < N; i++)
+            {
+                y1 = f(t + dt, y1);
+                y[i] = y1 + dt * (1.5 * f(t + dt, y1) - 0.5 * f(t, y0));
+                t += dt;
+                y0 = y1;
+                y1 = y[i];
+            }
+            return y;
         }
 
         public static double[] ThirdOrder()
@@ -75,4 +93,4 @@ namespace MathNet.Numerics.OdeSolvers
             return null;
         }
     }
-}
+}

src/UnitTests/OdeSolvers/OdeSolverTest.cs

@@ -27,9 +27,9 @@
 // OTHER DEALINGS IN THE SOFTWARE.
 // </copyright>
 
+using MathNet.Numerics.OdeSolvers;
 using NUnit.Framework;
 using System;
-using MathNet.Numerics.OdeSolvers;
 using System.Linq;
 
 namespace MathNet.Numerics.UnitTests.OdeSolvers
@@ -60,6 +60,30 @@ namespace MathNet.Numerics.UnitTests.OdeSolvers
             }
             Assert.AreEqual(1, ratio, 0.01);// Check error convergence order
         }
+
+        [Test]
+        public void AB2Test()
+        {
+            Func<double, double, double> ode = (t, y) => t + 2 * y * t;
+            Func<double, double> sol = (t) => 0.5 * (Math.Exp(t * t) - 1);
+            double ratio = double.NaN;
+            double error = 0;
+            double oldError = 0;
+            for (int k = 0; k < 4; k++)
+            {
+                double y0 = 0;
+                int N = Convert.ToInt32(Math.Pow(2, k + 6));
+                double dt = 2.0 / (N - 1);
+                double[] y_t = AdamsBashforth.SecondOrder(y0, sol(dt), 0, 2, N, ode);
+                error = Math.Abs(sol(2) - y_t.Last());
+                if (oldError != 0)
+                    ratio = Math.Log(oldError / error, 2);
+                oldError = error;
+                Console.WriteLine(string.Format("{0}, {1}", error, ratio));
+            }
+            Assert.AreEqual(2, ratio, 0.01);// Check error convergence order
+        }
+
         /// <summary>
         /// Runge-Kutta second order method for first order ODE.
         /// </summary>