A New Approach to Modeling of Physical Systems

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A New Approach to Modeling of Physical Systems
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Distinct and Disconnected Fields of Simulations
and Modeling

• Time-stepped
• Heavy emphasis on spatial grid techniques
• Used for solving PDEs
• Outdated time advance schemes
• Event-stepped
• Used in operations research, war games, etc.
• Has not incorporated spatial grid techniques
• Has only been applied to systems with small
  number of events

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A Simple Test Problem

• Start with an initial distribution of entities
• The speed of each entity is a function of the
  total number of entities in each bin
• New entities are periodically injected into
  the simulation zone

Time-stepped Algorithm
Courant Condition Dt fastest speed/bin width
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Standard Recipe
1. Pick one physics model 2. Use space and time
differencing of the eqns a. introduce
spatial grid
b. Advance the equations based on a uniform time
step
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Problems With Time-stepped Algorithm

• Tracks movement of each entity at the same time
  resolution Dt, independent of its speed.
• Recalculates the entitys speed regardless of
  whether the number of entities in a given bin has
  changed.
• Lots of computations even if there is no
  congestion.

Alternative Event Based Modeling
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Current Computational Demands

• ? Multi-scale Coupled Systems
• - Spatial scales vary from centimeters to
  200 RE
• - Temporal scales vary from less than
  milliseconds to
• days
• - Kinetic effects have large-scale
  consequences
• ? Multi-physics
• - electron physics e.g., controls
  reconnection
• - ion physics e.g., dominates formation
  of boundaries and transport
• - global features and dynamics e.g.,
  extent of the magnetotail
• - coupling to the ionosphere

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Structured Mesh
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Why Is the Current State-of-the-art Not Adequate?
1. Time advance scheme is too inefficient -
Every cell is updated regardless of whether a
change has occurred - Time step is not
based on local conditions
2. Existing time advance schemes do not allow
code-code coupling
3. The curse of Courant condition -
physically the system should be stable if there
is no change regardless of how fast information
is propagating but numerically it is not
Simulation Model Resolution Memory Requirement (Bytes) Run Time / 1 hour of Magnetosphere Time
Full Particle (mi/me200) Electron Scales 1017 1.2x107 Years
Hybrid Ion Scales 3x1011 2 Months
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A New Approach
Time-stepped Sophisticated spatial mesh
techniques Event-stepped Uses irregularly
time-stamped events that only updates what needs
to be updated when it needs to be updated
Combine the spatial mesh techniques of
time-stepped with time advance method of
event-stepped
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Test Example I
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Test Example II
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Summary

• By combining techniques from two distinct fields,
  more efficient algorithms are obtained
• Application to plasmas demonstrate speed up of
  over a factor of 100 in 1D, with projected speed
  of 105 in 3D
• General enough to be applicable to other areas
  where time-stepped methodology are in use