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Question
I am working on the problem ,"Flux Cancellation and Coronal Mass Ejection".Please suggest me the required mathematical procedures and computer software for this problem.

Answer
Hello,

You should be able to find a wealth of background information, as well as insights for computations here:

http://sprg.ssl.berkeley.edu/cism/papers/linkerpub03.pdf


Obviously, the set of equations given (1)- (6) will have to be solved numerically, or via numerical simulation.

One numerical simulation method that might be used is called "FTCS" or the forward time centered space scheme. Let’s say you’re looking at the plasma diffusion equation:  @f/@t - φ [@^2f/@x^2 + @^f/@y^2 @^f/@z^2 ] = 0

Where @ denotes partial deriviative


Applying finite differences, the preceding partial differential equation simplified to 1D is converted to:

f(n+1)/j = f n/j + s (delta x)^2  L_xx  fn/j  where:

s = φ(delta t) /(delta x)^2

The "truncation error" is then given by:

E n/j =  φ(delta x)^2/ 2(s - 1/6) @^4f/ @x^4  + O[(delta x)^4]

Thus, at each stage the magnitude of the error resulting from truncation of  numerical values can be estimated.

Generalizing this same approach to 2-dimensions, yields:

f(n+1)/jk = f n/jk + s_x (delta x)^2  L_xx  fn/jk +  s_y (delta y)^2  L_yy  fn/jk

with standard errors:

s _x = φ_x(delta t) /(delta x)^2
s_y = φ_y(delta t) /(delta y)^2

and, of course, this can also be extended to 3 dimensions, but of course the simulation equations increase in complexity.  Discretization  of parameters via numerical simulation methods always introduces errors and there are various types of such error sources. Typical errors for a discretization are due to:

1- Truncation and rounding.

2- The discrete representation and the limited computer resources (memory and speed).

3- Accumulation of errors in a systematic manner for certain numerical methods.

The way to improve on the results is to reduce the respective error source. This can be done by:

1) Demanding better spatial and/or temporal resolution (smaller grid space, more finite elements, larger number of base functions, smaller time step)

2) A change to a better and more appropriate simulation method (there are many!)

Of course, doing solar simulations is even more difficult because: a) there are many more particles to deal with per cm^3 (more like 10,000,000 vs. 1 - 10) and b) there are stronger forces acting to cause magnetic diffusion and nonlinearity, as well as acting over much larger regions.

A key point of attack will hinge on what manner of differential equation one is dealing with.

Please be aware I am not definitively answering your question here, only providing clues for an approach to your own solutions. The main problem then is converting equations (1)-(6)in the paper (link above)  to numerical analogs you can use for numerical simulations, solutions.

I am not aware of any single computer program that does the sort of numerical solution or simulation you asked for. Generally, at least in the say I was doing this stuff, we used FORTRAN code to write our own numerical solutions, for the differential equations involved.

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Philip A. Stahl

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I specialize in stellar and solar astrophysics. Can answer questions pertaining to these areas, including: stellar structure and evolution, HR diagrams, binary systems, collapsars (black holes, neutron stars) stellar atmospheres and the spectroscopic analysis of stars – as well as the magnetohydrodynamics of sunspots and solar flares. Sorry – No homework problems done or research projects! I will provide hints on solutions. No nonsense questions accepted, i.e. pertaining to astrology, or 'UFOs' or overly speculative questions: 'traveling through or near black holes, worm holes, time travel etc. Absolutely NO questions based on the twaddle at this Canadian site: http://members.shaw.ca/warmbeach/FAQ.htm purporting to show a "new physics". Do not waste my time or yours by wasting bandwidith with reference to such bunkum.

Experience

Have constructed computerized stellar models; MHD research. Gave workshops in astrophysics (stellar spectroscopy, analysis) at Harry Bayley Observatory, Barbados. More than twenty years spent in solar physics research, including discovery of SID flares. Developed first ever consistent magnetic arcade model for solar flares incorporating energy dissipation and accumulation. Developed first ever loop-based solar flare model using double layers and incorporating cavity resonators. (Paper presented at Joint AGU/AAS Meeting in Baltimore, MD, May 1994)

Organizations
American Astronomical Society (Solar physics and Dynamical astronomy divisions), American Geophysical Union, American Mathematical Society, Intertel.

Publications
Papers appearing in Solar Physics, Journal of the Royal Astronomical Society of Canada, Journal of the Barbados Astronomical Society, Meudon Solar Flare Proceedings (Meudon, France). Books: 'Fundamentals of Solar Physics', 'Selected Analyses in Solar Flare Plasma Dynamics', 'Physics Notes for Advanced Level', 'Astronomy & Astrophysics: Notes, Problems and Solutions', 'Modern Physics: Notes, Problems and Solutions'

Education/Credentials
B.A. degree in Astronomy; M.Phil. degree in Physics - specializing in solar physics.

Awards and Honors
Postgraduate research award- Barbados government; Studentship Award in Solar Physics - American Astronomical Society. Barbados Astronomical Society award for service (1977-91) as Journal editor.

Past/Present Clients
Caribbean Examinations Council (as advisor, examiner), Barbados Astronomical Society (as Journal Editor 1977-91), Trinidad & Tobago Astronomical Society (as consultant on courses, methods of instruction, and guest speaker).

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