Runge-Kutta second-order methods

The numerical methods that are available in Stella are Euler s method, Runge-Kutta second order, or Runge-Kutta fourth order. One of the menu items allows the user to specify the length of simulation time, as well as the time increment dt, and the type of numerical method that is to be employed. 

There are several methods that we can use to increase the order of approximation of the integral in eqn. (8.72). Two of the most common higher order explicit methods are the Adams-Bashforth (AB2) and the Runge-Kutta of second and fourth order. The Adams-Bashforth is a second order method that uses a combination of the past value of the function, as in the explicit method depicted in Fig. 8.19, and an average of the past two values, similar to the Crank-Nicholson method depicted in Fig. 8.21, and written as... 

In 52 the author develops a symplectic exponentially fitted modified Runge-Kutta-Nystrom method. The method of development was based on the development of symplectic exponentially fitted modified Runge-Kutta-Ny-strom method by Simos and Vigo-Aguiar. The new method is a two-stage second-order method with FSAL-property (first step as last). 

The basic idea of the Runge-Kutta methods is illustrated through a simple second order method that consists of two steps. The integration method is constructed by making an explicit Euler-like trail step to the midpoint of the time interval, and then using the values of t and tp at the midpoint to make the real step across the whole time interval ... 

To understand the mathematical bases of the Runge-Kutta methods, it is useful to look at the simple case of an explicit second-order method applied to a single first-order equation ... 

In [211] the authors obtained a new embedded 4(3) pair explicit four-stage fourth-order Runge-Kutta-Nystrom (RKN) method to integrate second-order differential equations with oscillating solutions. The proposed method has high phase-lag order with small principal local truncation error coefficient. The authors given the stability analysis of the proposed method. Numerical comparisons of this new obtained method to problems with oscillating and/or periodical behavior of the solution show the efficiency of the method. 

The major computational effort in applying the Rxmge-Kutta methods occurs in the evaluation of /. For second-order methods, two functional evaluations per step are required, and for the fourth-order method, four evaluations per step are required. As a consequence, the lower-order methods with smaller step size may be less costly than the higher-order methods using a larger step size. Relationships between evaluations per step and the order of local tmncation error can be found in the literature [24]. Also available in the numerical analysis literature [15] are developments that combine the best features of the Euler and a second-order Runge-Kutta method. 

Different choices of b yield different second-order Runge-Kutta methods. If 6 = the method is called the improved Euler s or Heun s method. If 6 = 1, the method is called the improved polygon or modified Euler s method. As demonstrated by this development, Runge-Kutta methods are not unique since they involve the choice of an arbitrary constant. All second-order methods involve the evaluation of two slopes ki and A)2 (equation (3.1.38) and equation (3.1.39)) and the value of is a weighted average of these two slopes (equations (3.1.35) and (3.1.36)). 

Standard numerical methods such as second-order Runge-Kutta could be used, but a more effective approach is to expand the potential energy surface in equation (2) to second order and integrate the resulting expression from Xi to 3c,+i. This yields the local quadratic approximation (LQA) of Page and Mclver " which is an explicit second-order method. 

Runge-Kutta methods are explicit methods that use several function evaluations for each time step. Runge-Kutta methods are traditionally written for/(f, y). The first-order Runge-Kutta method is Euler s method. A second-order Runge-Kutta method is... 

This improved procedure is an example of the Runge-Kutta method of numerical integration. Because the derivative was evaluated at two points in the interval, this is called a second-order Runge-Kutta process. We chose to evaluate the mean derivative at points Pq and Pi, but because there is an infinite number of points in the interval, an infinite number of choices for the two points could have been made. In calculating the average for such choices appropriate weights must be assigned. 

Solve Eq. (3) to obtain B+1 using the second-order TVD-Runge-Kutta method presented as follows ... 

Solve the convection equation of high order (3rd order) essentially non-oscillatory (ENO) upwind scheme (Sussman et al., 1994) is used to calculate the convective term V V

velocity field P". The time advancement is accomplished using the second-order total variation diminishing (TVD) Runge-Kutta method (Chen and Fan, 2004). 

In ref. 143 the authors develop a third-order 3-stage diagonally implicit Runge-Kutta-Nystrom method embedded in fourth-order 4-stage for solving special second-order initial value problems. The obtained method has been developed in order to have minimal local truncation error as well as the last row of the coefficient matrix is equal to the vector output. The authors also study the stability of the method. The new proposed method is illustrated via a set of test problems. 

In ref 156 the author studies the stability properties of a family of exponentially fitted Runge-Kutta-Nystrom methods. More specifically the author investigates the P-stability which is a very important property usually required for the numerical solution of stiff oscillatory second-order initial value problems. In this paper P-stable exponentially fitted Runge-Kutta-Nystrom methods with arbitrary high order are developed. The results of this paper are proved based on a S5unmetry argument. 

In ref 161 the authors have produced exponentially-fitted BDF-Runge-Kutta type formulas (of second-, third- and fourth-order). The good behaviour of the new produced methods for stilf problems is proved. The stability regions of the new proposed methods have been examined. It is proved that the plots of their absolute stability regions include the whole of the negative real axis. Finally the author propose several procedures to find the parameter of the new methods. 

In ref. 162 the numerical integration of Hamiltonian systems is investigated. Trigonometrically fitted S5miplectic partinioned Runge-Kutta methods of second, third and fourth orders are obtained. The methods are tested on the numerical integration of the harmonic oscillator, the two body problem and an orbital problem studied by Stiefel and Bettis. 

In essence, the fourth order Runge-Kutta method performs four calculation steps for every time interval. In the solution by Euler s method, decreasing the time increment to 5 seconds, to perform four times as many calculation steps, still only reduces the error to 0,9% after 1 half-life. 

In Section 5 trigonometrically fitted Fifth algebraic order Runge-Kutta methods are presented. For these methods we present the construction and the error analysis from which one can see that the classical method is dependent on the third power of energy, the first exponentially-fitted methods is depended on a second power of the energy and finally the second exponentially-fitted methods is depended on first power of the energy. 

This is a second-order Runge-Kutta method (Finlayson, 1980), sometimes called the midpoint rule. The first step is an approximation of the solution halfway between the beginning and ending time, and the second step evaluates the right-hand side at that mid-point. The error goes as (At), which is much smaller than that achieved with the Euler method. The second-order Runge-Kutta methods (there are several) also have a stability limitation. 

---