Web Based Interface for Numerical Simulations of Nonlinear Evolution Equations

1
Web Based Interface for Numerical Simulations of
Nonlinear Evolution Equations

• Ryan N. Foster Thiab Taha
• Department of Computer Science
• The University of Georgia
• Spring 2007

2
Abstract

• In Computational Science and Parallel Computing
  research, model equations have been developed to
  assist in solving problems in science and
  engineering. Such equations have aided
  researchers in developing methods used in the
  study of weather prediction, optical fiber
  communication systems, water waves, etc.
• Often, it is the desire of many researchers
  to further develop numerical methods and make
  these model equations, their numerical
  simulations and plots accessible to users through
  the Internet.
• In this presentation, we present a web based
  graphical user interface for numerical simulation
  of nonlinear evolution equations such as the
  nonlinear Schrödinger (NLS), NLS (NLS) with
  periodic dispersion, and modified coupled NLS
  (CNLS) equations. Sequential and parallel
  algorithms for each equation were implemented on
  sequential and multiprocessor machines.

3
Outline

• Introduction
• Goal
• Model Equations and Methods
• gnuplot
• Equation Server
• Equation Server Architecture
• Equation Server Configuration
• Requirements
• Demo
• Conclusion

4
Introduction

• In Computational Science and Parallel Computing
  research, numerical methods have been developed
  to assist in solving problems in Science and
  Engineering.
• It is the desire of many researchers to further
  develop numerical methods and make these model
  equations, their numerical simulations and plots
  accessible to users through the Internet.
• Installing, setting up, and maintaining currently
  available web servers can be a complex and
  tedious task.

5
Goal

• Provide developers an easier alternative way to
  publish model equations and methods on the web
• Make numerical simulations and plots accessible
  to users through the internet.
• Implement without the use of existing web servers
  (Apache/Tomcat, Axis) and/or additional
  technology (PHP, JSP, Applets).

6
Model Equations and Methods

• Nonlinear evolution equations to be considered
  are
• nonlinear Schrödinger (NLS) type equation
• nonlinear Schrödinger (NLS) equation with
  periodic dispersion
• modified coupled nonlinear Schrödinger (CNLS)
  type equation

7
Nonlinear Schrödinger (NLS) Equation

• Consider the NLS equation
• where is a complex-valued function.
• The initial condition is
• where a constant.
• We assume that satisfies periodic
• boundary condition with period ,
  where
• is half the length of the interval.

8
Nonlinear Schrödinger (NLS) Equation (continued)

• When the -P, P is normalized to ,
  the NLS equation becomes
• The interval can be divided into N equal
  subintervals
• The equation is then solved using the
  first-order, second-order, or fourth-order
  split-step Fourier methods1.

9
Nonlinear Schrödinger (NLS) Equation (continued)

• Using the first-order Fourier method, the
  solution may be advanced from time t to the next
  time-level t?t by the following steps
• Advance the solution using only the nonlinear
  part.
• through
• Advance the solution according to the linear
  part
• by means of computing

10
Nonlinear Schrödinger (NLS) Equation (continued)

• followed by
• and
• where ?t denotes the time step, is the
  discrete
• Fourier transform and is its inverse.

11
Nonlinear Schrödinger (NLS) Equation (continued)

• Using the second-order Fourier method, the
  solution may be advanced from time t to the next
  time-level t?t by the following steps
• Advance the solution using only the nonlinear
  part.
• through
• Advance the solution according to the linear
  part
• by means of the discrete Fourier transforms

12
Nonlinear Schrödinger (NLS) Equation continued

• Advance the solution using the nonlinear part
• through

13
Nonlinear Schrödinger (NLS) Equation (continued)

• Using the fourth-order Fourier method, the
  solution may be advanced from time t to the next
  time-level t?t by the following steps
• Advance the solution using the second-order
  split-step Fourier method described with
• Advance the solution in time from t ??t to
  t(1-?)?t by the second-order split-step Fourier
  method.
• Advance in time from t(1-?)?t to t?t by the
  second-order split-step Fourier method

14
Parallel Algorithms for Solving the Nonlinear
Schrödinger (NLS) Equation

• Steps to solve the NLS equation on a parallel
  system
• Let A, of size N, be the array that holds the
  approximate solution to u at time t.
• The array A is distributed among P processors.
  Processor n, 0 n P - 1, contains array
  elements AnN/P to A(n1)N/(P-1).
• Each of the P processor works on its own
  sub-arrays independently without communicating
  with others.

15
Parallel Algorithms for Solving the Nonlinear
Schrödinger (NLS) Equation (continued)

• The FFTWs MPI library routines are used to
  implement the parallel discrete Fourier
  transforms to parallelize the computations of
• and .

16
NLSE with periodic dispersion

• Consider the NLS equation of the form
• where D(t) represents dispersion, given by the
  following periodic function
• and g(t) relates to effective nonlinearity, which
  is given by the periodic function

17
NLSE with periodic dispersion (continued)

• In the numerical experiments in 4
• , , ? 0.8, the map
  period
• the damping coefficient G 4,
• and the amplifier spacing
• with periodic boundary condition
• -20, 20, and
• initial conditions of the form
• t 0, amplitude A 1, velocity O 2, initial
  position , and phase .

18
Modified Coupled Nonlinear Schrödinger (CNLS)
Equation

• Consider this CNLS equation
• where

U
,
,
,
,
,
,
,
,
g(z)
,
M is an integer and is a uniform random
variable in
19
Modified Coupled Nonlinear Schrödinger (CNLS)
Equation (continued)

• The system in (1) can be written as a coupled
  system in the form

20
Modified Coupled Nonlinear Schrödinger (CNLS)
Equation (continued)

• Initial Conditions
• ,
• , parameters n (solitons),
• , ,
• (parameters for the dispersion map,
• with )

21
Modified Coupled Nonlinear Schrödinger (CNLS)
Equation continued

• As for values to use initially, take
• d(z)1 (constant dispersion),
• Gamma 10, z_a 0.1,
• z_p 0.01,
• N1, A1, Omega0, T0, Phi0.
• propagate up to z_max 20.

22
Modified Coupled Nonlinear Schrödinger (CNLS)
Equation continued

• Ismail and Taha (to be submitted) developed the
  following Crank Nicholson method for solving the
  CNLS equation 6
• where

23
Modified Coupled Nonlinear Schrödinger (CNLS)
Equation continued

• with the Crank Nicolson scheme

24
Modified Coupled Nonlinear Schrödinger (CNLS)
Equation (continued)
25
Modified Coupled Nonlinear Schrödinger (CNLS)
Equation (continued)

• where

26
Modified Coupled Nonlinear Schrödinger (CNLS)
Equation (continued)

• They use the boundary conditions
• where s0, 1 at the boundaries i.e. at (m1,N).
• By separating real and imaginary parts and
  assuming

27
Modified Coupled Nonlinear Schrödinger (CNLS)
Equation (continued)

• one can get the following system
• where

28
gnuplot

• gnuplot is an interactive data and function
  plotting utility.
• Features
• Titling, labeling, and two and three-dimensional
  plotting capability.
• Additional plotting tools and numerical features
• A data file can serve as input.
• Plotting capability through the use of command
  line.
• Available free online.

29
gnuplot continued
gnuplot demo
http//www.gnuplot.info
30
Steps

• Provide model equations to users through a web
  browser.
• Allow user to select an equation and method.
• Obtain input from the user for the selected
  method.
• Execute the selected equation and method to
  obtain simulation results.
• Return simulation results to the user in data and
  plot format.

31
1) Provide model equations to users through a web
browser.
32
2) Allow user to select an equation and method.
33
3) Obtain input from the user for the selected
method.
34
4) Execute the selected equation to obtain
simulation results.
35
5) Return simulation results to user in data and
plot format.
36
Parallel Implementation

• For parallel algorithms, the same concept
  applies
• Provide model equation to users through a web
  browser.
• Allow user to select the model equation and the
  parallel method.
• Obtain input from the user for the selected
  method.
• Execute the selected parallel method to obtain
  simulation results.
• Return simulation results to the user in a
  readable data and plot format.

37
Equation Server Architecture
38
Equation Server Architecture continued
Server(s) Machines where Equation Solver is
installed
Equation Solver HTML(s)
Client(s) Web Browser
equation input solver
LAN/WAN
Equation Solver executables GNUPLOT
numerical method solver
Equation Solver Result(s)
equation result solver
39
Equation Server Configuration

• Equation Server simplifies the complex task of
  configuring existing web servers by reducing all
  configurations into one simple configuration
  file.
• The configure file specifies
• which ports to run on
• URL of Equation solvers
• Time to remove results from the system
• Available equations and input requirements

40
Equation Server Configuration
41
Requirements

• C compiler with a socket library (socket.h).
• gnuplot, the interactive data and function
  plotting utility.
• Machine with system-call capability.

42
Demo

• The Equation Server is currently running on three
  machines. Two of which, are high performance
  machines (machines with the capability to run
  parallel codes on multiple processors).
• atlas (1 dedicated processor)
• http//atlas.cs.uga.edu7334
• taha (4 dedicated processors)
• http//taha.cs.uga.edu7334
• altix (8 dedicated processors)
• http//altix.rcc.uga.edu7101

43
Conclusion

• Several Numerical Methods for solving nonlinear
  evolution equations were presented. These
  methods include Finite Difference and Split Step
  Fourier Methods.
• A web based interface Equation Server was
  presented, that gives researchers the ability to
  provide user access to model equations and
  methods. Users can simulate these equations to
  obtain simulation results through the Internet.

44
References

• 1 XU, X., Taha, T., 2003, Parallel Split-Step
  Fourier Methods for Nonlinear Schrodinger-Type
  Equations, Journal of Mathematical Modeling and
  Algorithms 2 185-201, 2003.
• 2 Taha, T., 1994, Numerical simulations of the
  complex modified Korteweg-de Vries equation,
  Mathematics and Computers in Simulation, 37
  (1994) 461-467.
• 3 Foster, R. N., 2007, Web Based Interface for
  Numerical Simulations of Nonlinear Evolution
  Equations, Thesis (MS). The University of
  Georgia.
• 4 Liu, R., 2001. Numerical and Parallel
  Algorithms for the CMKDV Equation, Thesis (MS).
  The University of Georgia.
• 5 gnuplot homepage, http//www.gnuplot.info
• 6 Taha, T. R. 2006, Parallel Numerical Methods
  for Solving Nonlinear Evolution Equations,
  Department of Computer Science.