Chapter 6: effect of Discritization

Question 1

The effect of the various discretizations in space and time may be studied systematically in the context of a

Answer 1

linearized model.

Question 2

The effect of the various discretizations in space and time may be studied systematically in the context of a linearized model.

 With such a model, it is possible to determine

Answer 2

analytical solutions both for the system of equations under consideration and for the system obtained after discretization

Question 3

The shallow water barotropic model is used as a

Answer 3

a study tool

Question 4

The shallow water barotropic model is used as a study tool, as it has as solutions the two types of waves described by the primitive equation models:

Answer 4

• slow (Rossby) waves associated with advection terms and
• fast inertia-gravity waves associated with the Coriolis terms and the adaptation terms (pressure force in the equations of motion and divergence in the continuity equation).

Question 5

The equations of the linearized shallow water model are obtained from

Answer 5

the corresponding nonlinear equations by using the small perturbation method.

Question 6

The equations of the linearized shallow water model are obtained from the corresponding nonlinear equations by using the small perturbation method.

 This consists in

Answer 6

writing the model variables as the sum of a time-independent basic state and of perturbations evolving over time.

Question 7

When using the spectral method, it can be considered that the

Answer 7

horizontal derivatives are evaluated exactly and that no error is made therefore (except for computer accuracy).

Question 8

When using the spectral method, it can be considered that the horizontal derivatives are evaluated exactly and that no error is made therefore (except for computer accuracy).

 Where finite differences are used on

Answer 8

a grid

Question 9

Where finite differences are used on a grid, the horizontal derivatives are

Answer 9

not calculated exactly and a discretization error is introduced

Question 10

Where finite differences are used on a grid, the horizontal derivatives are not calculated exactly and a discretization error is introduced, whose effects on ……………. must be calculated

Answer 10

wave propagation must be calculated.

Question 11

Effects of Horizontal Discretization

Answer 11

For this, the advection and adaptation terms of the linearized shallow water equations are discretized on various types of grids (A-D).

Question 12

Types of Grids

The A-type grid

Answer 12

is the simplest that can be imagined, with all the variables being expressed at the same place.

Question 13

Types of Grids

In B-type grid

Answer 13

the velocities u and v are calculated at the locations other than that of geopotential o|

Question 14

Types of Grids

The C-type grid

Answer 14

is characterized by the fact that the velocities u and v together with the geopotential o| are evaluated at different points.

Question 15

Types of Grids

D-type grid

Answer 15

The idea of D-type grid is to switch around the position of the variables on passing from one time step to the next.

Question 16

The idea of D-type grid is to switch around the position of the variables on passing from one time step to the next.

Answer 16

For even time steps the position of the variables is as with the C grid, except that the positions of velocities u and v are reversed.

• For odd time steps, all the variables switch position, the u′s take the places of the v ‘s, the v ‘s of the u′s and the geopotential o| is now calculated in the center of the grid.

Question 17

Three cases of discretization are distinguished for the ………… terms

Answer 17

advection

Question 18

Three cases of discretization are distinguished for the advection terms

Answer 18

• central differences on the A, B, or C grids
• discretization on the D staggered grid
• central differences of 4th-order accuracy on the A grid

Question 19

Results indicate that the effects of discretization appear for

Answer 19

the shorter waves only and become negligible when λ = 16Δx.

Question 20

Discretization of 4th-order accuracy has a clear advantage, although this is offset by

Answer 20

the greater volume of computation required.

Question 21

………… -grid gives better accuracy than using the ……………….

Answer 21

D-grid gives better accuracy than using the A, B, and C grids.

Question 22

Four types of discretization are distinguished for the ……….. terms:

Answer 22

adaptation

Question 23

Four types of discretization are distinguished for the adaptation terms:

Answer 23

• central differences on the A grid
• discretization on the B grid
• central differences on the C or D’ grids,
• 4th-order accuracy central differences on the A grid

Question 24

Results show that the effect of spatial discretization is fairly similar to

Answer 24

what is observed for slow waves.

Question 25

It is observed that the C or D grids provide greater accuracy than

Answer 25

the B grid, which in turn provides greater accuracy than the A grid.

Question 26

It is observed that the C or D grids provide greater accuracy than the B grid, which in turn provides greater accuracy than the A grid.

 Here again it is seen that at the cost of additional computation, the

Answer 26

4th-order accuracy discretization outclasses the 2nd-order accuracy discretizations.

Question 27

To study the properties of time discretization only, we assume that

Answer 27

the horizontal derivatives are computed exactly by the spectral method.

Question 28

Euler scheme

Answer 28

It can be shown that the Euler scheme, while decelerating waves, increases their amplitude from one time step to the next, the most prominent effect being for fast inertia-gravity waves.

Question 29

It can be shown that the Euler scheme, while decelerating waves, increases their amplitude from one time step to the next, the most prominent effect being for fast inertia-gravity waves.

• Therefore, the Euler scheme is

Answer 29

unconditionally unstable; its use in numerical prediction is reserved strictly to the first time step.

Question 30

Therefore, the Euler scheme is unconditionally unstable; its use in numerical prediction is reserved strictly to the first time step.

• The faster the …………………. the ……………………..

Answer 30

inertia-gravity waves, the more prominent the effect.

Question 31

The centered explicit scheme is

Answer 31

conditionally stable

Question 32

The centered explicit scheme is conditionally stable, the constraint on the …………………… depend essentially on ………………………………………

Answer 32

time step (CFL condition) depend essentially on the speed of the inertia-gravity waves.

Question 33

The centered explicit scheme is conditionally stable, the constraint on the time step (CFL condition) depend essentially on the speed of the inertia-gravity waves. When it is stable, it

Answer 33

accelerates the waves slightly, but preserves their amplitude.

Question 34

The centered explicit scheme is conditionally stable, the constraint on the time step (CFL condition) depend essentially on the speed of the inertia-gravity waves. When it is stable, it accelerates the waves slightly, but preserves their amplitude.

 However, because it utilizes

Answer 34

three time levels

Question 35

However, because it utilizes three time levels, it introduces in addition to

Answer 35

physical solutions, spurious solutions known as computational solutions that need to be brought under control in a model when it is integrated.

Question 36

The centered semi-implicit scheme is implemented by

Answer 36

evaluating the time derivative in central differences, the advection, and Coriolis terms at time t and by calculating the adaptation terms as the arithmetic mean of their values at time levels t + Δt and t − Δt.

Question 37

The semi-implicit scheme is

Answer 37

conditionally stable,

Question 38

The semi-implicit scheme is conditionally stable, the constraint on

Answer 38

the time step arising essentially from the velocity of the basic wind.

Question 39

The semi-implicit scheme is conditionally stable, the constraint on the time step arising essentially from the velocity of the basic wind. When it is stable, it

Answer 39

slightly accelerates the speed of slow waves, but considerably reduces the speed of inertia-gravity waves while conserving their amplitude

Question 40

The semi-implicit scheme is conditionally stable, the constraint on the time step arising essentially from the velocity of the basic wind. When it is stable, it slightly accelerates the speed of slow waves, but considerably reduces the speed of inertia-gravity waves while conserving their amplitude.

 It also introduces,

Answer 40

spurious computational solutions

Question 41

Time filtering is used to

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attenuate (reduction of the amplitude) the decoupling between even time steps and odd time steps brought about by the existence of the computational solution in central difference schemes,

Question 42

Time filtering is used to attenuate (reduction of the amplitude) the decoupling between even time steps and odd time steps brought about by the existence of the computational solution in central difference schemes, which uses

Answer 42

the values of the variables at three successive time levels (e.g., semi-implicit centered schemes).

Question 43

In continuous filtering procedure, the values of the

Answer 43

variable at a time step are calculated using the filtered values of the previous time step successively.

Question 44

Suppose that we know the filtered values at time t − Δt, noted X (t − Δt), the values at time t + Δt are calculated first: the filtering is then applied to

Answer 44

the values at next time steps successively.

Question 45

the filtering is then applied to the values at next time steps successively.

 Introducing filtering gives a more

Answer 45

restrictive CFL stability condition

Question 46

Introducing filtering gives a more restrictive CFL stability condition. The higher the

Answer 46

wavenumber k (and therefore the shorter the wavelength), the more attenuated is the physical solution and the computational solution is itself attenuated much more than the physical solution.