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Fenster schliessen

Euler forward scheme

To define time schemes, we consider the equation:

\(\frac{du}{dt}=f(u,t)\;\;\;where\;\;u=u(t)\)

  1. Two-level-schemes: use timesteps n and n+1
    1. \(u^{n+1}=u^n+\int^{(n+1)\Delta t}_{n\Delta t}f(u,t)dt\)
    2. Euler or forward scheme:
      • \(u^{n+1}=u^n+\Delta tf^n\)

      • first order accurate scheme: \(O(\Delta t)\)

      • uncentred

      • explicit

Fenster schliessen

Backward Scheme 

To define time schemes, we consider the equation:

\(\frac{du}{dt}=f(u,t)\;\;\;where\;\;u=u(t)\)

  1. Two-level-schemes: use timesteps n and n+1
    1. \(u^{n+1}=u^n+\int^{(n+1)\Delta t}_{n\Delta t}f(u,t)dt\)
    2. Euler or forward scheme:
      • \(u^{n+1}=u^n+\Delta tf^n\)

      • first order accurate scheme: \(O(\Delta t)\)

      • uncentred

      • explicit

    3. Backward scheme:

      • \(u^{n+1}=u^n+\Delta tf^{n+1}\)

      • first order accurate scheme: \(O(\Delta t)\)

      • uncentred
      • implicit 
    4. Trapezoidal scheme
      • \(u^{n+1}=u^n+\frac{1}{2}\Delta t(f^n+f^{n+1})\)
      • second order accurate scheme: \(O((\Delta t)^2)\)

      • implicit
    5. Matsuno or Euler-backward scheme
      • to increase the accuracy: first an Euler forward time step, than a backward step
      • \(1.\;u^{n+1/2}=u^n+\Delta t f^n\\ 2.\;u^{n+1}=u^n+\Delta tf^{n+1/2}\)
      • explicit
      • first order accurate scheme: \(O(\Delta t)\)

    6. Heun scheme
      • first Euler forward, then Trapezoidal
      • \(1.\;U^{n+1*}=U^{n}+\Delta t*f^n\\2.\;U^{n+1}=U^n+\frac{1}{2}\Delta t(f^n+f^{n+1})\)
      • second order of accuracy
      • explicit 

 

  1. Three-level-schemes: use timesteps n-1, n, n+1
    1. \(u^{n+1}=u^{n-1}+\int^{(n+1)\Delta t}_{(n-1)\Delta t}f(u,t)dt\)
    2. leapfrog scheme
      • \(u^{n+1}=u^{n-1}+2\Delta tf^n\)
      • second order accurate scheme: \(O((\Delta t)^2)\)

      •  

        most widely used scheme in atmospheric and ocean models. 

Fenster schliessen

Trapezoidal Scheme 

To define time schemes, we consider the equation:

\(\frac{du}{dt}=f(u,t)\;\;\;where\;\;u=u(t)\)

  1. Two-level-schemes: use timesteps n and n+1
    1. \(u^{n+1}=u^n+\int^{(n+1)\Delta t}_{n\Delta t}f(u,t)dt\)
    2. Euler or forward scheme:
      • \(u^{n+1}=u^n+\Delta tf^n\)

      • first order accurate scheme: \(O(\Delta t)\)

      • uncentred

      • explicit

    3. Backward scheme:

      • \(u^{n+1}=u^n+\Delta tf^{n+1}\)

      • first order accurate scheme: \(O(\Delta t)\)

      • uncentred
      • implicit 
    4. Trapezoidal scheme
      • \(u^{n+1}=u^n+\frac{1}{2}\Delta t(f^n+f^{n+1})\)
      • second order accurate scheme: \(O((\Delta t)^2)\)

      • implicit
    5. Matsuno or Euler-backward scheme
      • to increase the accuracy: first an Euler forward time step, than a backward step
      • \(1.\;u^{n+1/2}=u^n+\Delta t f^n\\ 2.\;u^{n+1}=u^n+\Delta tf^{n+1/2}\)
      • explicit
      • first order accurate scheme: \(O(\Delta t)\)

    6. Heun scheme
      • first Euler forward, then Trapezoidal
      • \(1.\;U^{n+1*}=U^{n}+\Delta t*f^n\\2.\;U^{n+1}=U^n+\frac{1}{2}\Delta t(f^n+f^{n+1})\)
      • second order of accuracy
      • explicit 

 

  1. Three-level-schemes: use timesteps n-1, n, n+1
    1. \(u^{n+1}=u^{n-1}+\int^{(n+1)\Delta t}_{(n-1)\Delta t}f(u,t)dt\)
    2. leapfrog scheme
      • \(u^{n+1}=u^{n-1}+2\Delta tf^n\)
      • second order accurate scheme: \(O((\Delta t)^2)\)

      •  

        most widely used scheme in atmospheric and ocean models. 

Fenster schliessen

Matsuno or Euler Backward Scheme 

To define time schemes, we consider the equation:

\(\frac{du}{dt}=f(u,t)\;\;\;where\;\;u=u(t)\)

  1. Two-level-schemes: use timesteps n and n+1
    1. \(u^{n+1}=u^n+\int^{(n+1)\Delta t}_{n\Delta t}f(u,t)dt\)
    2. Euler or forward scheme:
      • \(u^{n+1}=u^n+\Delta tf^n\)

      • first order accurate scheme: \(O(\Delta t)\)

      • uncentred

      • explicit

    3. Backward scheme:

      • \(u^{n+1}=u^n+\Delta tf^{n+1}\)

      • first order accurate scheme: \(O(\Delta t)\)

      • uncentred
      • implicit 
    4. Trapezoidal scheme
      • \(u^{n+1}=u^n+\frac{1}{2}\Delta t(f^n+f^{n+1})\)
      • second order accurate scheme: \(O((\Delta t)^2)\)

      • implicit
    5. Matsuno or Euler-backward scheme
      • to increase the accuracy: first an Euler forward time step, than a backward step
      • \(1.\;u^{n+1/2}=u^n+\Delta t f^n\\ 2.\;u^{n+1}=u^n+\Delta tf^{n+1/2}\)
      • explicit
      • first order accurate scheme: \(O(\Delta t)\)

    6. Heun scheme
      • first Euler forward, then Trapezoidal
      • \(1.\;U^{n+1*}=U^{n}+\Delta t*f^n\\2.\;U^{n+1}=U^n+\frac{1}{2}\Delta t(f^n+f^{n+1})\)
      • second order of accuracy
      • explicit 

 

  1. Three-level-schemes: use timesteps n-1, n, n+1
    1. \(u^{n+1}=u^{n-1}+\int^{(n+1)\Delta t}_{(n-1)\Delta t}f(u,t)dt\)
    2. leapfrog scheme
      • \(u^{n+1}=u^{n-1}+2\Delta tf^n\)
      • second order accurate scheme: \(O((\Delta t)^2)\)

      •  

        most widely used scheme in atmospheric and ocean models. 

Fenster schliessen

Heun Scheme 

To define time schemes, we consider the equation:

\(\frac{du}{dt}=f(u,t)\;\;\;where\;\;u=u(t)\)

  1. Two-level-schemes: use timesteps n and n+1
    1. \(u^{n+1}=u^n+\int^{(n+1)\Delta t}_{n\Delta t}f(u,t)dt\)
    2. Euler or forward scheme:
      • \(u^{n+1}=u^n+\Delta tf^n\)

      • first order accurate scheme: \(O(\Delta t)\)

      • uncentred

      • explicit

    3. Backward scheme:

      • \(u^{n+1}=u^n+\Delta tf^{n+1}\)

      • first order accurate scheme: \(O(\Delta t)\)

      • uncentred
      • implicit 
    4. Trapezoidal scheme
      • \(u^{n+1}=u^n+\frac{1}{2}\Delta t(f^n+f^{n+1})\)
      • second order accurate scheme: \(O((\Delta t)^2)\)

      • implicit
    5. Matsuno or Euler-backward scheme
      • to increase the accuracy: first an Euler forward time step, than a backward step
      • \(1.\;u^{n+1/2}=u^n+\Delta t f^n\\ 2.\;u^{n+1}=u^n+\Delta tf^{n+1/2}\)
      • explicit
      • first order accurate scheme: \(O(\Delta t)\)

    6. Heun scheme
      • first Euler forward, then Trapezoidal
      • \(1.\;U^{n+1*}=U^{n}+\Delta t*f^n\\2.\;U^{n+1}=U^n+\frac{1}{2}\Delta t(f^n+f^{n+1})\)
      • second order of accuracy
      • explicit 

 

  1. Three-level-schemes: use timesteps n-1, n, n+1
    1. \(u^{n+1}=u^{n-1}+\int^{(n+1)\Delta t}_{(n-1)\Delta t}f(u,t)dt\)
    2. leapfrog scheme
      • \(u^{n+1}=u^{n-1}+2\Delta tf^n\)
      • second order accurate scheme: \(O((\Delta t)^2)\)

      •  

        most widely used scheme in atmospheric and ocean models. 

Fenster schliessen

leap-frog scheme

To define time schemes, we consider the equation:

\(\frac{du}{dt}=f(u,t)\;\;\;where\;\;u=u(t)\)

  1. Two-level-schemes: use timesteps n and n+1
    1. \(u^{n+1}=u^n+\int^{(n+1)\Delta t}_{n\Delta t}f(u,t)dt\)
    2. Euler or forward scheme:
      • \(u^{n+1}=u^n+\Delta tf^n\)

      • first order accurate scheme: \(O(\Delta t)\)

      • uncentred

      • explicit

    3. Backward scheme:

      • \(u^{n+1}=u^n+\Delta tf^{n+1}\)

      • first order accurate scheme: \(O(\Delta t)\)

      • uncentred
      • implicit 
    4. Trapezoidal scheme
      • \(u^{n+1}=u^n+\frac{1}{2}\Delta t(f^n+f^{n+1})\)
      • second order accurate scheme: \(O((\Delta t)^2)\)

      • implicit
    5. Matsuno or Euler-backward scheme
      • to increase the accuracy: first an Euler forward time step, than a backward step
      • \(1.\;u^{n+1/2}=u^n+\Delta t f^n\\ 2.\;u^{n+1}=u^n+\Delta tf^{n+1/2}\)
      • explicit
      • first order accurate scheme: \(O(\Delta t)\)

    6. Heun scheme
      • first Euler forward, then Trapezoidal
      • \(1.\;U^{n+1*}=U^{n}+\Delta t*f^n\\2.\;U^{n+1}=U^n+\frac{1}{2}\Delta t(f^n+f^{n+1})\)
      • second order of accuracy
      • explicit 

 

  1. Three-level-schemes: use timesteps n-1, n, n+1
    1. \(u^{n+1}=u^{n-1}+\int^{(n+1)\Delta t}_{(n-1)\Delta t}f(u,t)dt\)
    2. leapfrog scheme
      • \(u^{n+1}=u^{n-1}+2\Delta tf^n\)
      • second order accurate scheme: \(O((\Delta t)^2)\)

      •  

        most widely used scheme in atmospheric and ocean models. 

Fenster schliessen

What is a staggered grid and what are the advantages? Explain it with the gravity waves. 

Lizenzierung: Keine Angabe

The one-dimensional gravity waves: 

\(\frac{\partial u}{\partial t}=-g\frac{\partial h}{\partial x},\;\;\frac{\partial h}{\partial t}=-H\frac{\partial u}{\partial x}\)

The differential/difference equations for the one-dimensional case are: 

\(\frac{u_j}{\partial t}=-g\frac{h_{j+1}-h_{j-1}}{2\Delta x},\;\;\frac{\partial h_j}{\partial t}=-H\frac{u_{j+1}-u_{j-1}}{2\Delta x}\)

If we use for that a non-staggered grid (like the A-grid), than we will have two elementary subgrids, with the solution on one of these subgrids beeing completely decoupled from the other. Thus it would be better to calculate only one of these solutions, that is, to use a staggered grid (like the C-grid). 

This reduces computation time by a fator of two and the truncation error stays the same. £££

Fenster schliessen

What grid is the most used for the linear shallow water equations? Explain in words.

  • We have shown, that the C-grid is the best choose for gravity waves. The error for the numerical solution can be decreased through the win in computation time. 
  • Although, wheres the space derivatives are straight forward for the C-grid, the Corilis term is complicated to calculate. 
  • To decide wheter which grid is to choose, we investigate the effect of the space distribution of dependent variables on the dispersive properties of the gravity-inertia waves. This will be done using the simplest centered approximations for the space derivatives, leaving the time derivatives in their differential form. That is the differential-diffrence equation.

The frequency equation for the one-dimesnional system (when wave soltions were plugged in) is: 

\((\frac{\vartheta}{f})^2=1+\frac{gH}{f^2}k^2\)

with the Rossby Radius of deformation is: 

\(R=\frac{\sqrt{gH}}{f}\)

It follows that the frquency of the gravity-inertia waves is monotonically increasing function of k. Therefore the group velocity is never zero. This is very important for the geostrophic adjustment process, as it precludes a local accumulation of wave energy. 

We now look at the effect of the finite differencing in space in this case. Obtaining again the frequency equation, we find that the non-dimensional frequency \(\vartheta/f\) depend on the two paramters: \(k\Delta x\) and \(R\Delta x\)

For the one-dimensional case C and B have also a monotinically increasing frequency as a function of wavenumber. Although the two-grid wave has a group velocity of zero. 

In the two-deminsional case, only the C-grid as no negative group velocities for all kombination of wavenumbers l and k. Although if the Rossby Radius is smaller than a half grid (\(\sqrt{gH}/f<\Delta x/2\)) than the group velocity is everywhere negative. However, this does not happen for typical grid sizes used in atmospheric models and thus the C-grid is the best choice to simulate the geostrophic adjustment process.