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The Leapfrog method for the solution of Ordinary Differential Equation initial value problems has been historically popular for several reasons. The method has second order accuracy, requires only one function evaluation per time step, and is non-dissipative. Despite the mentioned attractive properties, the method has some unfavorable stability properties. The absolute stability region of the method is only an interval located on the imaginary axis, rather than a region in the complex plane. The method is only weakly stable and thus exhibits computational instability in long time integrations over intervals of finite length. In this work, the use of filters is examined for the purposes of both controlling the weak instability and also enlarging the size of the absolute stability region of the method.

In this work the stability properties of a well known linear multistep method (LMM) are examined. The method is based on a three point polynomial finite difference approximation of the first derivative. The Leapfrog method (also known as the explicit midpoint rule or Nyström’s method) for the first order Ordinary Differential Equation (ODE) initial value problem (IVP)

is a simple three-level scheme with second order accuracy. The leapfrog scheme has a long history in applications. The applications include general oceanic circulation models and atmospheric models that go back at least as far as the pioneering work of Richardson [

The Leapfrog method is efficient as it only requires one function evaluation per time step, with a function evaluation referring to the function F in Equation (1). The leapfrog scheme has the desirable property of being non-dissipative, but it suffers from phase errors. The most serious drawback of the scheme is its well documented [

In the next section LMMs and their stability properties are reviewed. Then three and five point filters are developed and it is shown that a five-point symmetric filter can be effectively used to control the weak instability that is inherent to the leapfrog method. After the five-point symmetric is shown to be effective, ways to incorporate the filter into a leapfrog time differencing schemes are examined. The application of the filter leads to new LMMs as well as various filter and restart algorithms. The ideal application of the filter will control the weak instability as well as retain the favorable features of the method: efficiency, second order accuracy, an absolute stability region with imaginary axis converge, and will be non-dissipative. Finally, the conclusions are illustrated with numerical examples.

A general s-step linear multistep method for the numerical solution of the ODE IVP (1) is of the form

where

The stability of ODE methods is a relatively new concept in numerical methods and was rarely if ever used before the 1950’s [

The stability properties of LMMs are examined by applying the method to the ODE

where

where the

of the LMM (2). For a consistent method one of the terms, say

・ A method is stable (also called zero stable) if

・ A weakly stable method has all roots of

・ A method for which a root of

A stronger definition of stability can be made by considering non-zero

where

of the LMM (2). The method is absolutely stable for a particular value of z if the zeros of (6) satisfy the root condition. The set of all numbers z in the complex plane for which the zeros of (6) satisfy the root condition is called the absolute stability region,

that is the ratio of the characteristic polynomials of the method. For z in the stability region, the numerical solutions exhibit the same asymptotic behavior as does the exact solution of (3). For a linear system of ODEs, a necessary condition for absolute stability is that all of the scaled (by k) eigenvalues of the coefficient matrix of the system must lie in the stability region. For nonlinear systems, the system must be linearized and the scaled eigenvalues of the Jacobian matrix of the system must be analyzed at each

The leapfrog method for the system (1) is

The method is constructed by replacing the time derivative in (1) by the second order accurate central difference approximation

of the first derivative. The method needs two time levels to start,

The characteristic polynomials of the leapfrog method are

Obviously the roots of

and the absolute stability region only consists of the interval

It is shown in [

has an asymptotic expansion of the form

The physical mode

consists of the exact solution

contributes a non-physical oscillatory factor to the solution which grows in size with increasing t. References [

In order to damp the computational mode while retaining the second order accurate approximation of the physical mode, a filter of the form

is considered where

and

Thus, applying the filter (17) to the physical mode (15) results in

and applying the filter to the computational mode gives

Equation (18) suggests that in order for the physical mode to be kept unchanged up to the order

In a similar manner Equation (19) reveals that in order for the computational mode to be reduced by a factor of

First, filters of length three are considered. A three point symmetric filter is of the form

The filter coefficients can be found by equating the coefficients of (22) with the coefficients from Equation (20) and Equation (21). This can be done by setting

and

Additionally

where on the second line

and equating the coefficients of s gives

Similarly

where

The linear system consisting of Equations (26), (27), and (29) has the solution

Thus, the three-point symmetric filter is

or equivalently

Three point filters alter the accurate physical mode by a factor of k and also damp the computational mode by a factor of k.

The coefficients of non-symmetric filters

The consideration of a five point filter presents two more degrees of freedom that can be used to develop a filter that alters the physical mode by one less power of k and that damps the computational mode by one more power of k in comparison to a three point filter. That is, in the asymptotic expansion of the physical mode

A five point symmetric filter is of the form

In this case the first few terms in the expansions of (20) and (21) respectively are

and

Additionally

−1 | 2 | 3 | - | - | |

- | 1 | 2 | 1 | - | |

- | - | 3 | 2 | −1 |

where this time

Equating the constant terms (32) and (34) results in the equation

and then equating the coefficients of s gives

and finally equating the coefficients of

Similarly

where

and

The linear system consisting of Equations (36)-(38), (40), and (41) has the solution

or

The coefficients of non-symmetric filters

The advantages of the five point filter when compared to the three point filter are now clear. The five point filter retains the second order accuracy of the leapfrog method and damps the computational mode by a factor of

3 | −4 | −6 | 12 | 11 | - | - | - | - | |

- | −1 | 0 | 6 | 8 | 3 | - | - | - | |

- | - | −1 | 4 | 10 | 4 | −1 | - | - | |

- | - | - | 3 | 8 | 6 | 0 | −1 | - | |

- | - | - | - | 11 | 12 | −6 | −4 | 3 |

Incorporating the filter in each leapfrog step by averaging

or

This approach has the advantage that the result may be analyzed within the LMM framework. The characteristic

polynomials of the method are

polynomial has

Averaging

The roots of the first characteristic polynomial of the method are

A strong point of the leapfrog method, especially in the method of lines integration of wave propagation type PDEs, is that it is non-dissipative. The phase and amplitude errors of the two new LMMs are compared with the phase and amplitude errors of the leapfrog method in

In this section, four filter and restart algorithms are explored. For each option the effects on accuracy and stability properties are examined. The options include:

・ Method 1 (M1). Filter after every step.

・ Method 2 (M2). Filter followed by Euler restart every N steps.

・ Method 3 (M3). Alternative starting/restarting-algorithm 1.

・ Method 4 (M4). Alternative starting/restarting-algorithm 2.

To allow for the absolute stability region to be examined, all four algorithms have a start up procedure following by multiple steps of leapfrog and/or filtering. Then after a fixed number, N, of steps the algorithm is restarted with the start up procedure. In each of the four algorithms

The first approach uses an explicit Euler step (11) to calculate the first time level. Each subsequent step takes one leapfrog step to the time level n plus two additional leapfrog steps to calculate time levels

The absolute stability region of the M1 approach with

The absolute stability regions of methods M1, M2, M2, and M4 are found in the following manner. For example, the stability region of M1 with

and

The absolute stability region contains all the

Method M2 approximates the first time level with Euler’s method (11) and then advances in time with

The M2 approach remains second order accurate and requires

The absolute stability region of Euler’s method is a circle of radius one that is centered at

The following starting procedure

is suggested for minimizing this effect. To advance from the initial condition to time level one, M smaller substeps are taken consisting of one Euler step followed by

Algorithm M4 uses the same starting procedure (46) as does M3 but it does not terminate and restart after filtering at time step N. Instead time levels

covering approximately the interval

The purpose of the first example is to demonstrate the ability of the proposed methods to control the computational mode that is introduced by the leapfrog weak instability. The example considers the nonlinear IVP

which has the exact solution

This example numerically calculates the order of convergence of the leapfrog method and the seven filtered methods. The test Equation (3) is used with

leapfrog | 4.2e−6 | 3.0 | |

2.9e−5 | 0 | 0 | |

4.2e−6 | 3.3e−16 | 3.3e−16 | |

M1 | 1.3e−6 | 2.2e−16 | 2.2e−16 |

M2 | 7.8e−5 | 0 | 0 |

M3 | 1.7e−7 | 0 | 0 |

M4 | 2.6e−6 | 0 | 0 |

4.5e−5 | 2.2e−16 | 2.2e−16 |

M2 methods are only first order accurate while the LMM5, M3, and M4 methods retain the second order accuracy of the leapfrog method. In this example, the three second order accurate filtered methods are about a decimal place more accurate for a given value of k than is the leapfrog method.

The purpose of the third example is to give an illustration of the effects of the dispersion and dissipation properties of the two LMM described in Section 2 and to illustrate the effectiveness of the M4 algorithm as a method of lines integrator of hyperbolic PDEs. A model hyperbolic PDE problem is the advection equation

The domain for the problem is the interval

The space derivative of (48) is discretized with the Fourier Pseudospectral method [

remains to be advanced in time where D is the

Runs with two different time step sizes are made. The first with

Discretized advection-diffusion operators do not have purely imaginary eigenvalues and thus the leapfrog method is not an appropriate method of lines integrator for this type of problem. Advection-diffusion problems in which the advection term is dominant will have differentiation matrices with eigenvalues that have negative real parts that are of small to moderate size. Algorithm M4 can be effectively applied in such a problem.

leapfrog | 1.5e−2 | 1.5e−2 | 2.3e−1 |

5.7e−2 | 3.3e−1 | 7.3e−1 | |

1.4e−2 | 1.5e−2 | 2.2e−1 | |

M4 | 2.8e−3 | 2.8e−2 | 2.5e−1 |

As an example, the advection diffusion equation

where

is used and the problem is advanced in time to

Filters of length three and five have been examined for the purpose of controlling the weak instability of the leapfrog method. Filtering the leapfrog method with a three point filter reduces the accuracy of the overall scheme to first order. If a five point filter is used, it is possible for the method to retain the second order accuracy as well as damp the computational mode by a factor of

leapfrog | 1.2e−2 | 7.9e−3 | 1.1e−1 |

3.9e−2 | 2.5e−1 | 6.7e−1 | |

1.5e−2 | 1.0e−2 | 8.5e−2 | |

M4 | 1.1e−3 | 1.1e−2 | 1.2e−1 |

of the leapfrog method (9) results in a new second order accurate LMM (43) which is zero stable and that has an absolute stability region that includes both coverage of a portion of the imaginary axis as well as a region in the left half plane. Of the four filter and restart algorithms described, algorithm M4 is the most attractive. The M4 method, which restarts after every 21 time steps, has an absolute stability region that includes nearly as much of the imaginary axis as does the leapfrog method as well as includes a region in the left half plane. This allows the method to be an effective method of lines integrator for pure advection and advection dominated problems as is illustrated in the examples. In addition to the second order accuracy, the method also retains the computational efficiency of the leapfrog method. Method M4 is a potential replacement for existing leapfrog type time differencing schemes that are used in geophysical fluid dynamics codes.

Ari Aluthge,Scott A. Sarra,Roger Estep, (2016) Filtered Leapfrog Time Integration with Enhanced Stability Properties. Journal of Applied Mathematics and Physics,04,1354-1370. doi: 10.4236/jamp.2016.47145