ConvectionDiffusion Analyze based on Hermite Interpolation

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Convection-Diffusion Analyze based on Hermite
Interpolation

• Part II Results and Discussion
• By Yifei Dai and Luling Yang

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Compare Hermit and Lagrange

• Both Lagrange and Hermit interpolation can give
  reasonable solution.
• We noticed that when D0, Lagrange quadratic
  interpolation gave a better solution than Hermit
  interpolation.

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Peclet Number and Courant Number

• Peclet Number
• Courant Number
• viewed as a guide to minimizing numerical
  dispersion and oscillation
• For different scheme, Peclet number has different
  criteria.

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Peclet Number
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Peclet Number
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Peclet Number
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Courant Number
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Courant Number
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Courant Number
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Time Discretization Schemes
Implicit fraction 0 fully explicit
1 fully implicit
0.5 Crank-Nicolson
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Fully Explicit Scheme

• System matrix is symmetric
• System Matrix M
• Strict constrain on time step to keep
• stable solution.

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Fully Explicit Scheme
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Fully Explicit Scheme
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Fully Explicit Scheme
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Fully Implicit Scheme

• Implicit fraction1
• System matrix is unsymmetrical
• Contain convection matrix which is
• not symmetric
• No constrain for stable solution
• -We found for implicit fractiongt0.5,this
  will always be true.
• Smaller time step will result in more accurate
  solution.

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Fully Implicit Scheme
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Fully Implicit Scheme
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Fully Implicit Scheme
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Lumping of Mass Matrix M

• Diagonalization for simplification the problem to
  increase efficiency
• Only useful for fully explicit scheme
• Will decrease the accuracy of the solution.

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Lumping of Mass Matrix M
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Lumping of Mass Matrix M

• Lumping will increase efficiency for fully
  explicit scheme
• Strict constrain for stability exists in fully
  explicit scheme
• The effect of lumping and stability constrain
  will counteract each other.
• Accuracy decrease observed.

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Lumping of Mass Matrix M
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Lumping of Mass Matrix M

• We have not observed any increment of efficiency
  for implicit scheme
• The simplification caused by lumping will
  decrease the accuracy of solution.

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Conclusion

• Peclet and Courant number criteria provide the
  necessary conditions for the mesh design and
  selection of time steps
• Fully implicit scheme has higher efficiency but
  strict constrain of stability
• We prefer partial implicit scheme instead of
  lumping to increase efficiency.

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Future Work

• Analyze the problem with only fixed Ess. B.C.
• Analyze higher order Hermite interpolation

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Thank you!