Literature Study

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Literature Study
Literature Study
Jeroen Wille
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Structure

• Introduction
• Scalar advection equation
• Discontinuous Galerkin
• Conclusions and further research

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Introduction

• Two models
• Continuous
• Discontinuous
• Focus on numerical methods

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Scalar advection equation

• 1-dimensional
• SUPG
• Perturbation
• Variable velocity
• 2-dimensional
• SUPG
• Perturbation

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Scalar advection equation1D

• Equation
• Weak formulation
• Implicit Euler

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Scalar advection equation1D results continuous
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Scalar advection equation1D results discontinuous
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Scalar advection equation1D Perturbation

• Perturbation added to equation
• Neumann boundary condition needed at outflow
  boundary
• Extra term to stiffness matrix
• Extra term to right hand side

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Scalar advection equation1D Perturbation results
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Scalar advection equation1D SUPG

• Extra term p added to test function
• p is constant per element
• p is discontinuous over the elements
• Extra term to mass- and stiffnessmatrix

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Scalar advection equation1D SUPG results
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Scalar advection equation1D variable velocity

• v no longer fixed, but v v(x,t) defined by
• gradient of v equals zero, so v is constant in
  space
• Per time step, extra system solved to determine
  p(1)

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Scalar advection equation1D variable velocity
results
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Scalar advection equation2D

• Equation
• Weak formulation

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Scalar advection equation2D results continuous
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Scalar advection equation2D results discontinuous
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Scalar advection equation2D Perturbation

• Perturbation added to equation
• Neumann boundary condition needed at outflow
  boundary
• Extra term to stiffness matrix
• Extra term to right hand side

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Scalar advection equation2D Perturbation results
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Scalar advection equation2D SUPG

• Extra term p added to test function
• p is constant per element
• p is discontinuous over the elements
• Extra term to mass- and stiffnessmatrix

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Scalar advection equation2D SUPG results
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Discontinuous Galerkin

• Approximate u per cell by linear combination of
  polynomials
• Use legendre-polynomials as basis functions
• Use upwind flux

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Shock detection

• Detector
• Near discontinuity, IO(1)
• Otherwise I O(h(k2))
• So

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Limiting

• Use minmod function
• Limit in each cell from highest order till zero,
  or stop when coefficient does not change

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Overview
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Discontinuous Galerkinresults discontinuous
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Discontinuous Galerkinresults continuous
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Conclusions and further research

• FEM known in 2D, DG not
• Slope detector sometimes wrongly indicates
• Look at water flow model
• Apply numerical methods
• Obtain realistic data for this model
• Improve the model