Numerical Methods I, How to Solve Problem on a computer
Instructor: Prof. Varley Tuesdays and Thursdays 7:00-8:15PM
This course is taught in an electronic classroom room 1000, Lab B North Bldg (Macintosh) using Mathematica, as the programming and graphing language.
Textbook: Computational Physics 2nd edition by Nicholas Giordano and H. Nakanishi. (Prentice Hall, New Jersey). Chapters 1-4 and appendices A, B, D, and E are studied in PHYS385/685. The remainder of the book is studied in PHYS 485/695.
Much of physics is described by differential equations and it is therefore important that the student understand how to solve differential equations. Only a few kinds of differential equations can be solved exactly for simple models, and for real world problems, approximation methods or numerical methods must be used. It used to be that physics was broadly divided into theory and experiment but since about 1945 a new area called "computational physics" or numerical simulations has become important.
Numerical simulations uses computational tools on computers to solve physics problems in a manner that borrows from both theory and experiment. The actual theory used on the computer is either classical or quantum physics but the problem is explored in the manner of an experimentalist. Specifically a parameter is varied in the problem and the effect of that variation is observed in the computer output instead of an experiment. The computer simulation is a sort of "computer experiment". The methods of solving differential equations on a computer start with the Euler method and progress through Runge-Kutta.
The course will start with solving simple radioactive decay problems and heating/cooling problems so as to build confidence in the numerical methods of solution. Projectile problems with air resistance are studied and this is important as air resistance plays an important role in most real situations. Theoretical courses often neglect air resistance as it make the calculation too difficult but computational methods are very easy.
The mass on a spring problem is studied not just when the spring obeys the Hook's linear force law, but also when non-linear effects are important. Again numerical methods treats these nonlinear problems quite easily while the corresponding theoretical methods can be quite complex. Various realistic models of the solar system will be studied with computational methods and this is to be contrasted with the theoretical methods which have difficulty much beyond the two body problem. Finally, the various aspects of chaos or random that appear in nonlinear problems (like the "Butterfly effect") will be examined.
Quite often in physics various integrals appear as part of the solution and it is important to be able to "perform" these integrals. Mathematica can do a huge number of integrals symbolically and the student will learn how to use this feature for use in theoretical physics courses and research. However, many integrals that appear in physics can only be done numerically and various methods of efficiently, performing integrals numerically are discussed including Newton-Cotes, trapezoid rule, and Gauss method. Also, discussed are various methods of doing numerical derivatives as appear in the laboratory etc. Additionally various techniques for fitting experimental data to a function will be discussed including least square fitting using not just linear functions but polynomial, exponential, Sine functions etc.
Additionally, root finding of polynomial and transcendental equations is useful and will be taught using for example Newton-Raphson and secant methods
Pre-req: PHYS121 or 120 and two semesters of calculus. MATH 254 (ordinary differential equations) is NOT a pre-req. MATH 254 is not necessary for the student since the numerical techniques for solving differential equations taught in PHYS 385/685 are quite different from the techniques taught in MATH 254.
GRADING: There will be two midterm exams and a final exam. The highest of the two midterm exams counts 30% toward the final course grade and the final exam counts 40% toward the course grade.
Important Note: Work submitted by students on exams is required to be their own work and copying from others on exams (including take home exams) is consider plagiarism and subject to Hunter College rules of penalties of paralogism. The exams will begin in class and then continue on as take home exams. Homeworks assignments (workshops) are an important part of the course as there are eighteen workshop modules posted on the web (roughly one workshop per two lectures).
There will be time set aside in class for students to work on the workshop. Students are encouraged to work with a partner although each student is expected to turn in their own workshop done in their own way (that is, students may NOT copy workshops and hand them in). The workshops count 30% toward the course grade. Attendance in class is required and is part of the workshop grade.
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