Runge Kutta method

It is an accurate method of time integration and is explicit in nature. Table 3.15 summarises this method. This method of higher order is a robust algorithm used to solve non-linear equations but may have problems which have discontinuous coefficients which take place spatially. Some coefficients change discontinu-ously as the load increases. Strain localisation will be difficult to produce in numerical simulations. Non-linear equations have bifurcation. The non-linear equation of stochastic elasto-plasticity is more suitable for finding the most unstable solution compared with the non-linear equation of deterministic elasto-plasty since the coefficients change continuously. 

The Runge-Kutta method is widely used as a numerical method to solve differential equations. This method is more accurate than the improved Euler s method. This method computes the solution of the initial value problem. 

Process engineering and design using Visual Basic y = f x,y), y(Xo) = y at equidistance points 

This method is repeated hll the solution is reached. Example 1.27 

Solve Example 1.26 using the Runge-Kutta method. 

Similarly, other values can be calculated and tabulated as in Table 1.9 (calculated volume is 0.500007). 

The fourth-order Runge-Kutta method is a numerical method which, when the initial conditions are known (in particular that y(fo) = o). allows approximate solutions of differential equations of the type 

When h is small, higher order terms in h can be ignored. If we truncate after the term in h in the Taylor series expansion, we get the method of Euler, where 

From the value y and its derivative, this process can be continued through various steps to obtain the value of y +l. 

For a more precise treatment, we can truncate the expansion after the term in and obtain the second-order Runge-Kutta method. For this, we need to know the second derivative y (to), which can be estimated as a finite difference. 

Substituting this expression into the truncated Taylor series, we obtain 

In the fourth-order Runge-Kutta method, a set of formulas are used to calculate the model variables at the end of a time increment from their values at its beginning. Applied to the component material balance equations, the calculations proceed as follows  

The k factors in this equation are evaluated from the model variable values at the beginning of the time increment. For component material balances, 

Similar equations are written for the total material and energy balance equations. These equations are solved with the model equations to compute the variables at the end of each time increment. The method can be used off-line or on-line in a predictive mode. 

Using the simple tank model to illustrate the method, start by calculating the k factors  

This is followed by taking 90.937 kmol as the starting point for the next time increment, and completing the procedure as above. The results are tabulated herewith  

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Fig. 2. The time evolution of the total energy of four water molecules (potential-energy details are given in [48]) as propagated by the symplectic Verlet method (solid) and the nonsymplectic fourth-order Runge-Kutta method (dashed pattern) for Newtonian dynamics at two timestep values.

D. Janezic and B. Orel. Implicit Runge-Kutta method for molecular dynamics integration. J. Chem. Info. Comp. Sd., 33 252-257, 1993. 

Janezic, D., Orel, B. Implicit Runge-Kutta Method for Molecular Dynamics Integration. J. Chem. Inf. Comput. Sci. 33 (1993) 252-257 Janezic, D., Orel, B. Improvement of Methods for Molecular Dynamics Integration. Int. J. Quant. Chem. 51 (1994) 407-415... 

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... 

A popular fourth-order Runge-Kutta method is the Runge-Kutta-Feldberg formulas (Ref. Ill), which have the property that the method is fourth-order but achieves fifth-order accuracy. The popular integration package RKF45 is based on this method. 

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. 

More than two points can be used in the Runge-Kutta method, and the fourth-order Runge-Kutta integration is commonly employed. Obviously computers are... 

The Runge-Kutta method takes the weighted average of the slope at the left end point of the interval and at some intermediate point. This method can be extended to a fourth-order procedure with error 0 (Ax) and is given by... 

The simulated concentration-time values are displayed graphically in Fig. 5-1. These values could not have been obtained from analytical expressions. Most schemes that comprised a few steps can be handled by the Runge-Kutta methods, provided the time scales of the various steps are comparable. 

There are four equations in four dependent variables, a, d, T, and T. They can be integrated using the Runge-Kutta method as outlined in Appendix 2. Note that they are integrated in the reverse direction e.g., a = at) — similarly for 2 and in Equations (2.47). 

Pag), where y o mole fraction of A in bulk gas phase can be determined iteratively, yAi = mole fiaction of A in gas inlet. Equations (1) to (6) were solved using fourth order Runge-Kutta method [1, 8]. The value of enhancement factor, E, was predicted using equation of Van Krevelen and Hoftijzer [2]. 

Eqn. (5.5-101) solved numerically by Runge-Kutta method, least squares 0.0207 0.01... 

Figure 4 shows the output of this program, which consists of concentrations of o-methylol, p-methylol and methylene ether groups at various reaction times. Although many integration routines can be used in CSMP calculations, the variable interval Runge-Kutta method was used in this case since that is the option selected when no other method is specified. 

More complicated numerical methods, such as the Runge-Kutta method, yield more accurate solutions, and for precisely formulated problems requiring accurate solutions these methods are helpful. Examples of such problems are the evolution of planetary orbits or the propagation of seismic waves. But the more accurate numerical methods are much harder to understand and to implement than is the reverse Euler method. In the following chapters, therefore, I shall show the wide range of interesting environmental simulations that are possible with simple numerical methods. 

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). 

Both POLYMATH and CONSTANTINIDES use this method and also a fourth order Runge-Kutta method. Other methods are available in other software, but these two are adequate for the present book. 

The higher order ODEs are reduced to systems of first-order equations and solved by the Runge-Kutta method. The missing condition at the initial point is estimated until the condition at the other end is satisfied. After two trials, linear interpolation is applied after three or more, Lagrange interpolation is applied. 

Fixed step, 2nd-order, Runge-Kutta method (RK2). 

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