Additive RungeKutta Time Integration Of PDEs with GridInduced Stiffness

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Additive Runge-Kutta Time Integration Of PDEs
with Grid-Induced Stiffness
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What Problems Are We Trying To Solve?

• Far-field noise propagation
• Generally, PDEs with multiple spatial scales

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Far-field Noise Propagation Wind Turbine
Xu, Uzun, Hussaini, Kopriva, Harden (AIAA 2005)
used Euler equations with blades modeled by a
force term Mesh had 16200 elements, approx. 200
of which were small to resolve the tower Physical
frequency is typically 0.5Hz modeled frequency
was 100Hz Hence a realistic domain should be much
larger!
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Model Noise Propagation Using Euler Equations in
Conservation Form
where
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Relevant Spatial Scales In Far-Field Noise
Propagation

• Characteristic Lengths of Objects in Domain
• Wavelengths of Incoming Waves
• Wavelengths of Scattered Waves

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Multiple Spatial Scales Lead to Grid-Induced
Stiffness

• Complex geometric features gt small elements
• Spectral approximation gt incoming scattered
  waves resolved with only a few points per
  wavelength
• CFL condition on small elements gt explicit time
  step is smaller than what is needed to resolve
  waves.

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Sample Stiff Mesh Airfoil
A Joukowski airfoil (left) a zoom near the
airfoil (right).
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What is Additive Runge-Kutta?

• Single-step, multi-stage ODE solver for the
  problem

• Nonstiff summand is approximated with an explicit
  Runge-Kutta
• Stiff summand is approximated with an implicit
  Runge-Kutta method

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What is a Runge-Kutta Method?

• A Runge-Kutta method is a single-step
  multistage ODE solver for problems of the form
• where U can be a scalar or vector quantity.
• It is applied as

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The Butcher Array

• A Runge-Kutta (R-K) method is usually represented
  by its Butcher array The array consists of two
  vectors b,c, and a matrix A

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Explicit Runge-Kuttas (ERKs)

• Matrix A is lower triangular, zero diagonal
• Calculation of each stage relies only on
  the values of the previous stages.
• Stability region is always bounded
• stages order, with equality only for s lt 6

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(E)xplicit First-Stage, (S)ingly (D)iagonally
(I)mplicit (R)unge-(K)utta Methods
Butcher array has diagonal entries Therefore we
must solve for each stage value

• Advantages
• A-stability
• Explicit 1st stage allows stage-order 2
• Requires an implicit solve for only one variable
  at each stage,
• Allows for stiff accuracy and L-stability

• Disadvantages
• Considerably higher ops count at each stage
• Considerably higher storage requirement

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How ARK Is Used To Handle Grid-Induced Stiffness

• Large elements integrated explicitly
• Small elements integrated implicitly
• CFL condition on large elements used to set time
  step

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IMEX versus Explicit Integration

• Consider the linear advection equation with exact
  solution

• Ratio of largest element to smallest element
  (stiffness ratio) is 100

Data below are for solution after propagating 100
wavelengths. Avg. of GMRES iterations 50
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The Heisenberg Uncertainty Principle