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\bf{{\Large
A Numerical Methodology for Reducing Total Simulation
Error
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A. Louise Perkins \\
Naval Research Laboratory \\
Peter A. Orlin \\
Planning Systems Incorporated \\
Lucy F. Smedstad \\
Naval Research Laboratory \\
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\begin{abstract}
For more than three decades we have known that higher order accurate
schemes for evolution equations can be disappointing in practice.
This can be partly explained
by analyzing the types of error present.
Realistically initialized
formally time accurate studies,
based on reducing the numerical truncation error only, may be misleading.
The physically-based  mathematical approximations, used to
construct the governing equations for the simulation, may fail to adequately
model
long time dynamics.  Order of magnitude estimates of physical
parameters in the governing equations may even lead to failure
in short time.
This is because formal accuracy may be nullified by the
small errors in the mathematical formulation of the equation.
The Taylor Series-based numerical approximations,
when used with realistic (data-based) initial conditions,
lose
convergence.
As the
numerical grid approaches a continuum, the error bars on data
remain the same.  Hence the derivatives of
data values taken closer and closer together
approximate instrumental noise, not physics.
Also, the details of the simulation configuration may
introduce errors.

Taken together, the above family of errors
lead us to conclude that preparing a realistic model simulation should
consider some measure of the total simulation error, rather
than just focus on formal order of numerical truncation error.
Towards this end,
we define a total simulation error measure, and discuss
how it more accurately tracks the
performance of a realistic simulation (this is essentially
what is already done in numerical weather prediction).
We then show an example of a numerical approximation that has been
designed to
reduce total simulation error, and discuss the methodology
used in the design.

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