Local truncation error

The Gear Algorithm [15], based on the Adams formulas, adjusts both the order and mesh size to produce the desired local truncation error. BuUrsch and Sloer method [16, 22] is capable of producing accurate solutions using step sizes that arc much smaller than conventional methods. Packaged Fortran subroutines for both methods are available. 

To control the step size adaptively we need an estimate of the local truncation error. With the Runge - Kutta methods a good idea is to take each step twice, using formulas of different order, and judge the error from the deviation between the two predictions. Selecting the coefficients in (5.20) to give the same a j and d values in the two formulas at least for some of the internal function evaluations reduces the overhead in calculation. For example, 6 function evaluations are required with an appropriate pair of fourth-order and fifth-order formulas (ref. 5). 

High-level DAE software (e.g., Dassl) makes a time-step selection based on an estimate of the local truncation error, which depends on the difference between a predictor and a corrector step [13,46]. If the difference is too great, the time step is reduced. In the limit of At 0, the predictor is just the initial condition. For the simple linear problem illustrated here, the corrector will always converge to the correct solution y2 = 1, independent of the time step. However, if the initial condition is y2 1, then there is simply no time step for which the predictor and corrector values will be sufficiently close, and the error estimate will always fail. Based on this simple problem, it may seem like a straightforward task to build software that identifies and avoids the problem, and there is current research on the subject [13], The problem is that in highly nonlinear, coupled, problems the inconsistent initial conditions can be extremely difficult to identify and fix in a general way. 

The successful numerical solution of differential equations requires attention to two issues associated with error control through time step selection. One is accuracy and the other is stability. Accuracy requires a time step that is sufficiently small so that the numerical solution is close to the true solution. Numerical methods usually measure the accuracy in terms of the local truncation error, which depends on the details of the method and the time step. Stability requires a time step that is sufficiently small that numerical errors are damped, and not amplified. A problem is called stiff when the time step required to maintain stability is much smaller than that that would be required to deliver accuracy, if stability were not an issue. Generally speaking, implicit methods have very much better stability properties than explicit methods, and thus are much better suited to solving stiff problems. Since most chemical kinetic problems are stiff, implicit methods are usually the method of choice. 

Analysis of the algorithm requires understanding the behavior of local truncation error. We use the nomenclature y(tn) to mean the exact analytic solution evaluated at some time tn and yn to represent the numerical solution at tn. It is clear that the true analytic solution must satisfy the differential equation everywhere,... 

By adding the forward (explicit) finite-difference approximation to each side of this equation, we can identify both the explicit Euler algorithm and an expression for the local truncation error ... 

The first three terms represent the forward Euler algorithm operating on the exact solution, with the last term [in square brackets] providing a measure of the local truncation error. The local truncation error can be identified through a Taylor series expansion of the solution about the time tn ... 

Thus, considering only the linear term in the expansion, the local truncation error is h2... 

In addition to the restrictions that stability places on the step size hn, we also need to be concerned with how accuracy affects the choice of step size. Assume that the local accuracy is to be controlled to within a certain tolerance e and that accuracy can be estimated by the local truncation error. The time step must be chosen to keep a norm of the local truncation error below the tolerance, that is ... 

In regions where the solution varies slowly, accuracy considerations alone would permit a large time step. However, for a stiff problem where nearby solutions vary rapidly, stability demands a very small time step, even in regions where the solution is changing slowly and the local truncation error (accuracy) can be controlled easily with a large time step. 

The first three terms represent the implicit Euler algorithm and the remaining [bracketed] term represents the local truncation error. A Taylor series expansion about tn+ (in the negative t direction) yields an expression for y(t )... 

Vode, solves stiff systems of ordinary differential equations (ODE) using backward differentiation techniques [49]. It implements rigorous control of local truncation errors by automatic time-step selection. It delivers computational efficiency by automatically varying the integration order. 

The Local Truncation Error for the above differential method is given by L.T.E.(h) =... 

We now present the formulae of the Local Truncation Error (LTE) for the above methods. 

Introducing the expressions obtained in (24) into the Local Truncation Error of the methods mentioned above (see relations (15)-(23)), we obtain the following expressions (as polynomials of d) for Local Truncation Error of the methods. 

The local truncation error of this method is given by ... 

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. 

Error Analysis. In this paragraph we will study the behavior of the error and for this reason we have used the local truncation error (LTE), that is the difference between the theoretical and the approximate solution. We have studied the analytic form of the local truncation error for the three cases of... 

The local truncation error of the method (75) with coefficients given by (79)-(80) is give by ... 

---