62
2. Basic Finite-DifferenceMethods
Since Xcentered is real, it does not produce any change in the phase of the solution.
In the limit K ßt -+ 0,
showing that well-resolved oscillations undergo an O[(ßt)2] damping. The centered filter has the greatest impact on the most poorly resolved component of the
solution, the 2ßt oscillation. According to (2.49), each filter application reduces
the amplitude of the 2ßt wave by a factor of 1 - 4y. If y is specified to be
each filtering operation will completely eliminate the 2ßt oscillation.
Robert (1966) and Asselin (1972) suggested a scheme to control the leapfrog
computational mode by incorporating an approximate second-derivative time filter into the time integration cyc1e. They proposed following each leapfrog step
by the filtering operation
(2.50)
A filter parameter of y = 0.06 is typically used in global atmospheric models .
Values of y = 0.2 are common in convective cloud models; indeed, Schlesinger
et al. (1983) recommend choosing y in the range 0.25-0.3 for certain advectiondiffusion problems .
Ifthe Asselin-filtered leapfrog scheme is applied to the oscillation equation, the
amplification factor is determined by the simultaneous equations
(2.51)
(2.52)
A 2rpn-1 = rpn-I + 2iKßtArpn-l,
Arpn-I = Arpn-I + y (rpn-I _ 2Arpn-1 + A 2rpn-l) .
Under the assumption that (Arpn) = A(rpn), whose validity will be discussed
shortly, (2.52) may be written
(A - y)rpn-I = A ((l - 2y) + Ay) rpn-I .
(2.53)
Eliminating rpn-I between (2.51) and (2.53) yields
(2.54)
which reduces to the result for the standard leapfrog scheme when y = O. In the
limit of small K D.t, the amplification factor for the Asselin-filtered physical mode
become s
.
AAsselin-LF = 1 + IKßt -
(K ßt)2
4
+ O[(Kßt) ].
2(1 - y)
2. Basic Finite-DifferenceMethods
Since Xcentered is real, it does not produce any change in the phase of the solution.
In the limit K ßt -+ 0,
showing that well-resolved oscillations undergo an O[(ßt)2] damping. The centered filter has the greatest impact on the most poorly resolved component of the
solution, the 2ßt oscillation. According to (2.49), each filter application reduces
the amplitude of the 2ßt wave by a factor of 1 - 4y. If y is specified to be
each filtering operation will completely eliminate the 2ßt oscillation.
Robert (1966) and Asselin (1972) suggested a scheme to control the leapfrog
computational mode by incorporating an approximate second-derivative time filter into the time integration cyc1e. They proposed following each leapfrog step
by the filtering operation
(2.50)
A filter parameter of y = 0.06 is typically used in global atmospheric models .
Values of y = 0.2 are common in convective cloud models; indeed, Schlesinger
et al. (1983) recommend choosing y in the range 0.25-0.3 for certain advectiondiffusion problems .
Ifthe Asselin-filtered leapfrog scheme is applied to the oscillation equation, the
amplification factor is determined by the simultaneous equations
(2.51)
(2.52)
A 2rpn-1 = rpn-I + 2iKßtArpn-l,
Arpn-I = Arpn-I + y (rpn-I _ 2Arpn-1 + A 2rpn-l) .
Under the assumption that (Arpn) = A(rpn), whose validity will be discussed
shortly, (2.52) may be written
(A - y)rpn-I = A ((l - 2y) + Ay) rpn-I .
(2.53)
Eliminating rpn-I between (2.51) and (2.53) yields
(2.54)
which reduces to the result for the standard leapfrog scheme when y = O. In the
limit of small K D.t, the amplification factor for the Asselin-filtered physical mode
become s
.
AAsselin-LF = 1 + IKßt -
(K ßt)2
4
+ O[(Kßt) ].
2(1 - y)