1. Finite Difference Methods
18-20 September 2012

2. Thematic Outline Methods for computing temporal finite differences
Methods for computing spatial finite differences
Applications of grid staggering
Impacts of truncation error that arises from the finite difference approximations

3. Temporal Finite Differences
Note: the temporal differencing methods to be discussed apply to grid-point and spectral models.
Two types of time differencing schemes…
Explicit: all future time terms appear on one side of an equation
Implicit: future time terms appear on both sides of an equation
Hybrid explicit/implicit schemes also exist.
Explicit differences can be solved directly; implicit differences must be solved iteratively.

4. Explicit Temporal Differences
Two methods…
Single step (forward and centered differences)
Multiple step (predictor-corrector methods)
Single step example, centered difference:
where f is some dependent field (u, v, T, etc.) and F is the forcing on that field

5. Explicit Temporal Differences
Single step example, forward-in-time difference:
Let us consider these in the context of a real-world example, a one-dimensional advection equation…

6. Explicit Temporal Differences
Centered difference:
Forward-in-time difference:
Note: in both cases, we applied a second-order centered finite difference in space. We’ll discuss this method further shortly.

7. Explicit Temporal Differences
In predictor-corrector schemes, a prediction for the dependent field at some future time is first made using some finite difference approximation.
From this, the forcing at that future time is obtained, whether from finite difference or spectral methods.
That forcing is then used, often with a forward-in-time difference, to integrate to the next time.

8. Explicit Temporal Differences
Example: Runge-Kutta type scheme…
Predict:
With f*, obtain F*.
With fτ, obtain Fτ.
Correct:
Predictor-corrector methods provide superior accuracy at the expense of efficiency.

9. Explicit Temporal Differences
Issue: acoustic and other fast-moving waves
These waves require small time steps to maintain computational stability in the advection terms (CFL)
If not filtered, how to handle need for a small time step?
Potential method: split-explicit schemes
Small ∆t for terms representing fast-moving waves
Large ∆t for terms representing slower-moving waves
Reduces number of computations on short time steps
Commonly used with current mesoscale model systems

10. Implicit Temporal Differences
In explicit schemes, the forcing terms (spatial derivatives) are evaluated only at the initial time t.
In implicit schemes, these terms are evaluated at the future time t+1 and the initial time t.
In other words, they are valid at t+1/2.
Thus, terms at t+1 appear on both sides of the equation and the time derivative must be solved iteratively.

11. Implicit Temporal Differences
Let us again consider the example of a one-dimensional advection equation…
Explicit, forward-in-time approximation:
Instead of representing the spatial derivative exclusively at t=t, let us represent it as the average of its values at the initial (t) and future (t+1) times…

12. Implicit Temporal Differences
Spatial finite difference approximation:
If we substitute and apply a second-order centered spatial finite difference, this gives us equation (3.21),

13. Implicit Temporal Differences
Advantage: implicit schemes are typically unconditionally stable
CFL criterion for advection terms is thus not a concern
Enables us to use a longer time step to handle all waves
Disadvantage: iterative nature requires more computations per time step than explicit schemes
Longer time step savings < iterative computational expense

14. Implicit Temporal Differences
To alleviate these issues, semi-implicit methods may be used.
Meteorological terms: handled explicitly
Non-meteorological terms: handled implicitly
Alleviating the need for a small time step to maintain linear stability that arises with explicit methods.
A common, long time step may be used for all terms because implicit terms are stable for all time steps.
Advantage over split-explicit schemes.
Iterative computational expense is still present, however.

15. Spatial Differencing
We must first establish our frame of reference: fixed to the Earth or fixed to air parcels.
Thus far, we have considered a frame of reference tied to the Earth, an Eulerian reference frame.
All terms are defined and computed on a fixed grid.
Example: 1-D advection equation… ∆x ∆x
x-1 x
x+1

16. Spatial Differencing
But, as we’ve discussed before, the CFL criterion applies to the advection terms in this framework.
Can we utilize a reference frame that encompasses the advection terms and eliminates this issue?
Yes – if we use a reference frame tied to the motion (i.e., fixed to air parcels)!

17. Spatial Differencing
Such a reference frame is known as a Lagrangian reference frame.
Because Lagrangian methods follow air parcels, the spatial and vertical resolutions of the model are flow-dependent.
Finer resolution is found in convergent regions.
Coarser resolution is found in divergent regions.
This is not ideal.

18. Spatial Differencing Alternative: semi-Lagrangian methods
In a Lagrangian framework, only one set of parcels is followed, but is followed over all time steps.
In a semi-Lagrangian framework, a new set of parcels (defined by the chosen grid) is followed at each time step.
Parcels are followed utilizing parcel trajectories.
Either forward or backward trajectories may be used.
The spatial derivatives (advection terms) are implicitly accounted for by the along-trajectory evolution.
The bold & italic statement is important to note here. Our finite difference representations to come will look a lot like temporal differences, and yet they implicitly account for the spatial differences (advection terms) as well!

19. Semi-Lagrangian Differencing
Example: forward trajectories
Evenly-spaced grid (black) is defined at t=t.
Trajectories (red lines) are released from each grid point and followed forward in time to t=t+1 (red dots).
Non-advection terms in the primitive equations change the values of atmospheric fields along the trajectories.
The values of these fields at t=t+1 are then interpolated back to a uniform grid and the process repeated.
x-1 x
x+1

20. Semi-Lagrangian Differencing
Example: backward trajectories
Evenly-spaced grid (black) is defined at t=t+1.
Trajectories (red lines) are released from each grid point and followed backward in time to t=t (red dots).
Non-advection terms in the primitive equations change the values of atmospheric fields along the trajectories.
Need to interpolate from evenly-spaced grid valid at t=t to obtain this forcing.
But, no interpolation needed for t=t+1 – already on uniform grid!
x-1 x
x+1

21. Semi-Lagrangian Differencing
Backward trajectory methods are preferred.
Easier to interpolate from evenly-spaced grids to non-gridded trajectory locations than the inverse.
But, how can we represent the example on the previous slide numerically (and not just graphically)?

22. Semi-Lagrangian Differencing
Imagine a two-dimensional grid, as below, at t=t+1. Our parcel location at this time is given by (i,j).
(i,j)

23. Semi-Lagrangian Differencing
The backward trajectory from (i,j) is given by the red line. The parcel locations at t=t-1 (x) and t=t (+) are noted in blue. X +
(i,j)

24. Semi-Lagrangian Differencing
Let the location of X be and the location of + be X +
(i,j)

25. Semi-Lagrangian Differencing
Let the dependent variable in question be given by f and the forcing by F.
We want ft+1 and know ft-1 and ft by interpolation between grid points at the previous times.
F is computed locally using Eulerian finite difference methods and then interpolated between grid points.
The finite difference can then be explicit or implicit.

26. Semi-Lagrangian Differencing
Explicit: recall our centered-in-time difference...
The values of ft+1, ft-1, and Ft are taken at their locations along the trajectory, i.e.,

27. Semi-Lagrangian Differencing
Implicit: a generic example can be expressed as...
As for the explicit case, the values of ft+1, ft, Ft+1, and Ft are taken at their locations along the trajectory,

28. Semi-Lagrangian Differencing
Advantages of semi-Lagrangian schemes...
Loss of advection terms eliminates CFL criterion, enabling the usage of a longer time step.
Can be used for either grid point or spectral methods.
Can also be used with semi-implicit differencing schemes.
Minimizes non-linear instability (to be discussed soon).
Disadvantage of semi-Lagrangian schemes…
Not energy or mass conserving (a particular problem over long model integrations)

29. Grid Staggering
For some applications, it can be useful to define our dependent variables on two slightly offset grids.
This is known as grid staggering.
Both horizontal and vertical grid staggering are possible.
A typical staggering increment is ½ the grid interval.
Which variables apply to each grid depends upon the desired computational characteristics.

30. simple 1-D example for u and θ
Grid Staggering
simple 1-D example for u and θ
(a): must use θj+1 and θj-1 (over 2∆x) to compute a centered finite difference
(b): can use θj+½ and θj-½ (over ∆x) to compute a centered finite difference
Staggering halves the effective grid increment and minimizes the impacts of truncation error on the solution.

31. Grid Staggering
There are many examples of staggered grids in current NWP models.
Arakawa C-grid: mass variables staggered from kinematic variables; used in WRF-ARW model

32. Grid Staggering Other staggers…
Arakawa B-grid: akin to C grid, but with mass and kinematic variables staggered on grid centers and corners rather than grid centers and sides
Arakawa E-grid: akin to C grid, except rotated 45°
These grids are used by the MM5/NMM-B and WRF-NMM mesoscale models, respectively.

33. Grid Staggering
Grid staggering increases effective resolution but does not increase the number of grid points.
Increase in effective resolution requires a smaller time step to meet CFL criterion.
Only if Eulerian differencing methods are used, however.
Recall: as ∆x decreases, Courant number increases.
The benefit from reduced truncation error outweighs time step considerations, however.

34. Truncation Error
Many of the primitive equations are differential equations with partial derivative terms.
If we utilize grid point methods, these terms must be approximated with finite differences.
These approximations introduce error, however, depending upon their complexity.
This error is known as truncation error.

35. Finite Difference Approximations
First, however, it is useful to revisit how finite difference approximations are obtained.
Any function over an interval – i.e., a partial derivative – can be approximated by a Taylor series.
Generally, for a function in x,

36. Finite Difference Approximations
Expanded to several terms, we obtain (3.24),
Here, a is our point of interest and x is some point of reference. R is a residual term representing the error.

37. Finite Difference Approximations
This equation is used to obtain finite difference approximations for the function f.
The accuracy of each approximation is dependent upon the highest power/order terms retained in the equation.
Retain higher order terms: increased accuracy.
Keep only lower order terms: decreased accuracy.

38. Finite Difference Approximations
Example: first order / two point, forward in space
Let x = a+∆x and truncate 2nd order and higher terms…
Solve for the derivative…

39. Finite Difference Approximations
Example: first order / two point, backward in space
Let x = a-∆x and truncate 2nd order and higher terms…
Solve for the derivative…
First order schemes are typically not used, however, because they have large truncation errors.

40. Finite Difference Approximations
Example: second order / three point, centered
Let x = a+∆x and truncate 3nd order and higher terms…
Let x = a-∆x and truncate 3nd order and higher terms…

41. Finite Difference Approximations
Subtract the 2nd equation from the first…
Solve for the first derivative…

42. Finite Difference Approximations
A five point / 4th order approximation is given by…
where the two closest points on each side of a are used in the calculation of the derivative.
Any of these – or even higher-order – expressions may be used in numerical models to compute spatial derivatives in physical / grid-point space.

43. Truncation Error
Example: let f be a function for which the derivative is exactly known and compare that result to the finite difference approximations.
As in the text, let f = A cos kx, where k = 2π/L and L is the wavelength.
Analytic
Centered Difference

44. Truncation Error
The ratio of the approximation to the exact value gives the truncation error, where a value of 1 indicates no error.
As in the text, trig identities can be used to show:
Since k = 2π/L, this ratio varies with ∆x and L.

45. Truncation Error How does this ratio vary with ∆x and L?
If L is fixed (e.g., depicting some meteorological feature), as ∆x -> 0, ∆x/L -> 0.
As ∆x/L -> 0, k∆x -> 0.
By identity, as k∆x -> 0, sin (k∆x) -> k∆x.
Thus, for small ∆x (fine grid spacing), the ratio approaches 1 and the truncation error is small (i.e., the wave is well-resolved).

46. Truncation Error
If n = # of grid points, then L = n∆x.

47. Truncation Error As ∆x gets smaller, n gets larger (for fixed L).
Ratio asymptotes to 1 as n increases.

48. Truncation Error As L gets smaller, n gets smaller (for fixed ∆x).
Ratio asymptotes to 1 (0) for very large (small) waves.

49. Truncation Error
Let’s consider a practical example. Let f = A cos kx represent a ridge-trough pattern.
If L = 1000 km, what ∆x is needed for n = 10 (i.e., fairly small truncation error)?
Since L = n∆x, ∆x = L/n = 1000 km / 10 = 100 km.
Similarly, if L = 50 km (e.g., mesohigh-wake low), we need ∆x = 5 km for n = 10.

50. Truncation Error
Thus, to reasonably “resolve” a wave of length L, you need a horizontal grid spacing of at most L/10.
Ideally, you want an even-finer grid spacing to better resolve the wave and minimize truncation error.
Similar arguments can be made in the inverse to describe the wave(s) that can be resolved for a given horizontal grid spacing.

51. Grid Spacing Versus Resolution
Grid spacing and resolution are NOT interchangeable!
Resolution: defined by the wave that the model can resolve for a given grid spacing.
The resolution depends upon the grid spacing, yes, but also upon the accuracy of the finite difference approximation used (if using grid point methods).
Better approximation -> less error -> finer resolution

52. Truncation Error
Consider n=6…
4th order X X
2nd order

53. Truncation Error
The ratio does not approach 0.96 for the 2nd order scheme until beyond the limits of the chart.
Thus, if simpler finite difference methods are desired, many more grid points n (smaller ∆x) are/is needed to achieve an acceptable truncation error!

54. Truncation Error
With either 4th or 2nd order schemes, the error is still unacceptably large at small scales (n<5). We’ll discuss how we can deal with this later.
Errors end up manifest in all fields that evolve depending upon gradients…
Geostrophic wind (pressure gradient)
Advection terms (kinematic gradients)
Vertical motion from continuity (kinematic gradients)
…and grow upscale and with increasing time!

55. Truncation Error: Take-Home Points
Must consider the length of feature to be resolved.
Must consider the # of grid points (or grid spacing) needed to resolve that feature.
Must consider the error inherent to the desired finite difference approximation to be used.
Aside: this is partially why spectral modeling evolved, but that has its own unique issues that creep up…