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+PHYS 305 - Assignment #4
+Due: Friday, February 8th
+
+Make sure your name is listed as a comment at the beginning of all your work.
+
+Purpose: Illustrate the accuracy of the Differential Equations
+solvers. Study a Phase Space Portrait. Write an analysis tool.
+
+The one Gaussian potential hill solution
+========================================
+
+Understand and check your version of the code we did in class. Call
+this code one_gaussian_bump.c.  Make sure your version defaults to the
+following:
+
+* Height depth = 5, mass m = 1.0
+* E = 0.5
+* x = 3.5, v = −sqrt(2E/m)
+* tmin = 0.0, tmax = 7.0, dt = 0.001
+
+Make sure that the execution line modifiers, i.e., -d 0.001, allow you
+to reset the initial values of x and v, as well as dt (there could be
+more options).
+
+Accuracy - Order of the solver
+==============================
+
+Hint: Do not modify the code one_gaussian_bump.c in this section!
+
+You want to check the order of the differential equation solver. For
+this you need to find α in
+
+  error = dt^α
+
+based on the energy which ought to be a constant. Define
+
+  error = max((Ecalculated - Einitial)/Einitial)
+
+where the max means to find the maximum of the error over the entire
+time domain over which you integrate.
+
+* Write a small analysis program, error_analysis.c, which reads in
+  stdin from the stdout of one_gaussian_bump.c via a pipe
+
+    ./one_gaussian_bump | ./error_analysis
+
+  and computes error.
+
+* Write a shell script ( bash or tcsh ) to run the codes in the pipe
+
+    ./one_gaussian_bump -d0.001 | ./error_analysis
+
+  over a range in dt, [ 0.00004, 0.0002, 0.001, 0.005, 0.025 ]
+
+* Use the Euler, Mid-Point and RK4 (will be explained later in class)
+  methods in solving for the solution of the system of diff. eq.
+
+* Plot error for the three methods of solution versus dt in the
+  appropriate way to extract the power α.
+
+Phase Space Portrait
+====================
+
+Hint: Do not modify the code one_gaussian_bump.c in this section!
+
+* Use the script written above to run ./one_gausian_bump over a range
+  of initial energies, [ 0.1, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3,5, 4.0,
+  4.5, 4.9, 5.0, 5.1, 5.5 ]. Use the default dt = 0.001 and the RK4
+  method of solution.
+* Record the output data from all the runs in a single (big) file.
+* Plot all the trajectories, v versus x, at once in a single plot
+  (with dots).
+* Plot E versus x corresponding to the different initial energies at
+  once in a single plot.  Superpose a plot of the potential function
+  onto the latter (with dots).
+* Comment on what you see.