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Numerical Uncertainty Assessment. Reference : P.J. Roache, Verification and Validation in Computational Science and Engineering , Hermosa Press, Albuquerque (1998). Verification = Solving the equations right. OBJECTIVE.

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Numerical uncertainty assessment

Numerical Uncertainty Assessment

Reference: P.J. Roache, Verification and Validation in Computational Science and Engineering, Hermosa Press, Albuquerque (1998).

Verification solving the equations right
Verification = Solving the equations right


Show that the numerical solution converges to the exact solution of the differential equation as and .


Solve problem with different x and t and compare.

Validation solving the right equations
Validation = Solving the right equations

Show that the differential equations accurately model the physics of the problem.



Compare computational solution to experimental results.


  • Must always precede validation

  • Check that physical features of the solution are adequately resolved (boundary layers, crack tips, etc.)

  • Document how much solution changes when x and t are changed. Determine convergence rate.

  • Compare to exact solution of PDE

  • Compare solutions with two different numerical methods


  • Account for uncertainty in both computational and experimental results.

  • Important to validate using the features that are most sensitive to the model assumptions. (It is common that some features compare well to an experiment while other features of a numerical solution compare poorly.)

Example flow past a cylinder
Example: Flow Past a Cylinder


Color shows vorticity field (red – positive, blue – negative)

Comparison Parameters

Drag coefficient

Lift coefficient

Strouhal number

Verification compare solutions on different grids
Verification: Compare solutions on different grids

  • Fine Grid Computation: 21,580 points

  • Course Grid Computation: 10,790 points

Validation comparison to experimental results
Validation: Comparison to experimental results

Comparison of computational results with reference experimental data compiled

by Fleischmann and Sallet (1985).

Richardson extrapolation
Richardson Extrapolation

- A method to make a calculation more accurate

Let f be a numerical solution on a grid with step size h. Expand f about the exact solution to write

where g1, g2, g3, etc., are independent of h.

Richardson extrapolation1
Richardson Extrapolation

Consider a second-order calculation, such that g1 = 0. Perform calculations on two grids, with increment size h1and h2, such that



Richardson extrapolation2
Richardson Extrapolation

Taking h22(1) – h12(2) and solving for fexactgives


Hence, we can decrease the error in the numerical solution from O(h2) to O(h3) by forming the ratio above. For grid doubling (h2 = 2h1) we have

Grid convergence index gci
Grid Convergence Index (GCI)

- A method to quantify the error in a computation

Let p = order of computational method

r = grid refinement ratio

We can then write (3) as


Grid convergence index gci1
Grid Convergence Index (GCI)

Define the Grid Convergence Index (GCI) as the relative error, given by


Substituting this into (4) gives

GCI is a measure of the relative discretization error of the computed solution

Numerical dissipation
Numerical Dissipation

In approximation of convection on a fixed grid, it is typically necessary to introduce additional “artificial” or “numerical” dissipation in order to stabilize the convective transport term. This numerical dissipation can be either implicit in the differencing method or added explicitly to the computation.

Example lax wendroff method
Example: Lax-Wendroff Method

Problem: Develop a numerically stable method to solve the advection equation to second-order accuracy.


Lax wendroff method
Lax-Wendroff Method

Approach: Expand about

using a Taylor series with local error O(k3)


Differentiate (5) with respect to t to write


Lax wendroff method1
Lax-Wendroff Method

Substitute (7) into (6) and solve for


Plug (8) into (5) to get


FTCS Scheme


Numerical Dissipation Term


Effects of numerical dissipation
Effects of Numerical Dissipation

Causes regions of large  to expand in size and decrease in strength

Example for advection of a vortex in an inviscid fluid:


t = 0

t > 0

Most commercial codes have high numerical dissipation because it makes the computations very stable.

Methods for avoid numerical dissipation
Methods for Avoid Numerical Dissipation

  • Use Lagrangian method (no numerical dissipation)

  • Use high-order accurate method

  • Use local grid refinement in regions with large values of 

  • Use discretely-conservative method