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Bifocal approach to physics problems
What are numerical analysis??
● Consists of approximations to solve mathematical problems.
● Consists of approximations to solve mathematical problems.
● Examples:Solving ODEs to understand planetary motions.
● Consists of approximations to solve mathematical problems.
● Examples:● Solving ODEs to understand planetary
motions.● Numerical linear algebra for data analysis.
● Consists of approximations to solve mathematical problems.
● Examples:● Solving ODEs to understand planetary motions.
● Numerical linear algebra for data analysis.
● Markov chains,Stochastic DE to for bilogical systems
Lets not forget about analytical methods!!
Advantages of analytic methods
Advantages of analytic methods● Exact solutions.
Advantages of analytic methods● Exact solutions.● Helps understanding the physics behind the
problem.
Advantages of analytic methods● Exact solutions.● Helps understanding the physics behind the
problem.● Does not require heavy computation.
Numerical vs analytical method● Stochastic process cannot be very well understood by
analytical but numerical methods prove useful.
Numerical vs analytical method● Stochastic process cannot be very well understood by
analytical but numerical methods prove useful.
● Numerical solutions are difficult to interpret in contrast with analytical.
Numerical vs analytical method● Stochastic process cannot be very well understood by
analytical but numerical methods prove useful
● Numerical solutions are difficult to interpret in contrast with analytical.
● Numerical methods lead to difficulties with precision,singularities, stability.
● A major advantage of numerical method is that a numerical solution can be obtained for problems, where an analytical solution does not exist.
● A major advantage of numerical method is that a numerical solution can be obtained for problems, where an analytical solution does not exist.
● An additional advantage is, that a numerical method only uses evaluation of standard functions and the operations: addition, subtraction, multiplication and division.
Programming languages● Fortran: simple, old-fashioned, good at numerical
computation, good handling of vectors and matrices
● C: more powerful for strings/non-numerical data,
weaker than Fortran for numerical calculations, more
powerful control structures (relative to Fortran-77 but
not Fortran-90)
● – C++: object oriented approach, poorer numerical
performance relative to Fortran/C, industrial standard
?!?!● Numerically any differential equation can be solved!
● Can you state some examples ??
Types of numerical recipes● Finite Difference method.
Methods on a grid● Finite difference approximations for derivatives are
one of the simplest and of the oldest methods to solve differential equations.
RK 2,3,4
QUIZ TIME !!!!!!!
Have a 5 min break :)
What is bifocal approach ?
Types of numerical recipes● Finite Difference method.● Smooth Particle Hydrodynamics.
https://www.youtube.com/watch?v=M2Xqi2FBfiM
Basic Concept of SPH
To the board :)
Hydrodynamical simulations of cosmological structure formation
{Ref:GADGET (https://wwwmpa.mpa-garching.mpg.de/gadget/hydrosims/index.html)}
Types of numerical recipes● Finite Difference method.● Smooth Particle Hydrodynamics.● Finite Volume method.
So on!
Finite Volume Method● A finite difference method discretization is based upon the differential form
of the PDE to be solved. Each derivative is replaced with an approximate difference formula.
● Finite volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. Useful for problems with body-fitted coordinate systems.
