The Numerical Solution of Stochastic PDEs an application in hydrology

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The Numerical Solution ofStochastic PDEs -an
application in hydrology

• K. Burrage, T. Tianhai, P. Burrage
• Mathematics Department
• University of Queensland
• kb_at_maths.uq.edu.au

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Contents

• 1. The stochastic model
• 2. Method of Lines
• 3. Computing a stochastic process in time and
  space
• 4. Numerical methods and stiffness
• 5. Results and conclusions
• 6. Bibliography

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1. The stochastic model

• Modelling solute transport in an aquifer
• Darcys Law

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• Introduce the concept of stochastic velocity -
  describes motion of a tracer in a porous medium
• represents uncertainties due to
  pore structure.
• Stochastic flux - to model hydrodynamic
  dispersion
• is Diagonal diffusivity matrix,
  same form.

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• Simplification
• homogeneous and isotropic,
• constant, 1
  dimensional
• Approaches
• strong solutions - trajectories
• weak solutions - statistics
• distributions

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2. Method of Lines

• Uniform spatial mesh
• Second order differential - central
• First order - forward

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• Linear
• Stiff in deterministic and stochastic components

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• 3. Stochastic processes in
• time and space
• Consider stochastic process
• with mean 0, spatial covariance
• C is bounded, symmetric and p.d., with spectral
  decomposition

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• Kahunen-Loeve expansion
• Hydrological Example
  

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• Define the spatial correlation matrix
• compute Eq

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• Examples of spatial correlation

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4. Numerical methods and stiffness

• General SDE (Ito form)
• Euler methods - strong order 0.5

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• Apply to
• Balanced method - strong order 0.5

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• Choose such that
• has an inverse and
• Applied to Eq,
• The can be chosen appropriately for
  hydrological problem so that M is diagonal with
  diagonals gt 1.

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5. Numerical Results

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Bibliography

• T. Tianhai, K. Burrage and R. Volker (1999), The
  stochastic model and numerical simulations for
  solute transport in porous media.
• J.H. Cushman (1987), Development of stochastic
  partial differential equations for subsurface
  hydrology, Stoch. Hydrol. Hydraul. 1, 241-264.
• G. Dagan (1989), Flow and transport in porous
  formations, Springer, Berlin.
• Y. Rubin (1997), Transport of inert solutes by
  groundwater recent developments and current
  issues in subsurface flow and transport a
  stochastic approach, CU Press