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
Reference: P.J. Roache, Verification and Validation in Computational Science and Engineering, Hermosa Press, Albuquerque (1998).
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.
Show that the differential equations accurately model the physics of the problem.
Compare computational solution to experimental results.
Color shows vorticity field (red – positive, blue – negative)
Comparison of computational results with reference experimental data compiled
by Fleischmann and Sallet (1985).
- 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.
Consider a second-order calculation, such that g1 = 0. Perform calculations on two grids, with increment size h1and h2, such that
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
- 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
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
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.
Problem: Develop a numerically stable method to solve the advection equation to second-order accuracy.
Approach: Expand about
using a Taylor series with local error O(k3)
Differentiate (5) with respect to t to write
Substitute (7) into (6) and solve for
Plug (8) into (5) to get
Numerical Dissipation Term
(STABLE IF CFL # < 1)
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.