Midpoint rule

Improper integrals of the other types whose problems involve both limits are handled by open formulas that do not require the integrand to be evaluated at its endpoints. One such formula, the extended midpoint rule, is accurate to the same order as the extended trapezoidal rule and is used when the limits of integration are located halfway between tabulated abscissas ... 

We know from (3.13) in Chap. 3, how that same derivative approximation is of higher order 0(8t2) when applied at the midpoint, and this leads to the trapezium method or midpoint rule, in which we must find an expression for the right-hand side of (4.1) at time t + 6t. This can be approximated as the average of the values at both ends ... 

Bader, G. Deuflhard, P. A semi-implicit midpoint rule for stiff systems of ordinary differential equations. Numer. Math. 1983, 41, 373-398. 

The open formulas are not as accurate as the closed ones, hence most of the open formulas are rarely used. However, in engineering practice the simplest open formula, known as the midpoint rule, is frequently used as basis for the FVM. 

The midpoint rule is obtained by a polynomial of order zero which passes through the midpoint assuming that the function is constant over the integral of integration. The surface integral can thus be approximated as a product of the mean value over the surface and the cell face area [49]. The integrand at the cell face center is frequently used as an approximation to the mean value over the surface. In this case the surface integral is written as ... 

This integral approximation is of second order, provided that the value of / at location e is known. However, the value of / is generally not available at cell faces, hence they have to be obtained by interpolation from the node values. In order to preserve the second order accuracy of the midpoint rule approximation of the surface integral, the value of /e must be obtained with at least second order accuracy. 

Several terms in the transport equations require integration over the volume of a grid cell. The midpoint rule is again the simplest second order approximation available. The second-order approximation thus consists in replacing the volume integral by the product of the mean value and the GCV. The mean value is approximated as the value at the GCV center ... 

If the integral on the right side of (12.55) is estimated using the value of the integrand at the midpoint point of the time interval, we obtain the midpoint rule ... 

Depending on what further approximations are employed, the midpoint rule may give rise to single-step or multistep methods. 

For example, the second order (two-step) Leapfrog method frequently used in meteorology and oceanography can be deduced from the midpoint rule [66, 49, 158] ... 

The quadratic interpolation has a third order truncation error on both uniform and non-uniform grids [114, 49]. However, when this interpolation scheme is used in conjunction with the midpoint rule approximation of the surface integral, the overall approximation is still of second order accuracy (i.e., the accuracy of the quadrature approximation). Although the QUICK approximation is slightly more accurate than CDS, both schemes converge asymptotically in a second order manner and the difference are rarely large [49]. 

The the source terms are approximated by the midpoint rule, in which S is considered an average value representative for the whole grid cell volume. 

The the source terms are approximated by the midpoint rule, in which (S) is considered an average value representative for the whole grid cell volume. The derivatives are represented by an abbreviated Taylor series expansion, usually a central difference expansion of second order is employed. 

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. 

In section 9.2 we illustrated one explicit method, Euler s forward method. In the present section, we likewise used only one type of implicit method, based on the trapezoidal or midpoint rule. All our examples have used constant increments Af higher computational efficiency can oftenbe obtained by making the step size dependent on the magnitudes of the changes in the dependent variables. Still, these examples illustrate that, upon comparing equivalent implicit and explicit methods, the former usually allow larger step sizes for a given accuracy, or yield more accurate results for the same step size. On the other hand, implicit methods typically require considerably more initial effort to implement. 

The only open Newton-Cotes of practical interest is the midpoint rule ... 

If the parameters Aij, aij and Eij are all known, the initial concentrations and a temperature profile are given, the rate equations would predict the behaviour of the reaction. For very large systems a program LARKIN that integrates the, in general stiff, system of equations [27]. The initial value problems may be solved by routines like METANl [29] or SODEX [30, 31]. Both methods are based on a semi-implicit midpoint rule. 

With the help of a modified Crank-Nicol son scheme (midpoint rule) and... 

As a better approximation to the propagator (4.52) we can use a midpoint rule to estimate the integral in the exponential... 

The method is based on another time-marching scheme not mentioned in the above sections the leapfrog method [28, 29], also called the midpoint rule by Hairer and Wanner [6], using central differences. Equation (4.1) can be approximated as... 

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