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Computing Solutions of Ordinary Differential Equations
Published in Nayef Ghasem, Modeling and Simulation of Chemical Process Systems, 2018
The midpoint method uses the derivative at the starting point to approximate the solution at the midpoint. The midpoint method can be summarized as follows: The Euler method is used to estimate the solution at the midpoint.The value of the rate function f(x,y) at the midpoint is calculated.This value is used to estimate yi+1.
Differential Equations
Published in James P. Howard, Computational Methods for Numerical Analysis with R, 2017
Despite the importance of Runge–Kutta methods, they all share a common flaw around how many function evaluations are necessary to calculate each step. The Euler method requires a single function evaluation and the midpoint method requires two function evaluations for each step. As the Runge–Kutta methods increase in accuracy, more evaluations are required and rungekutta4 requires four function evaluations for each step. These add up quickly in applied problems and, for larger initial value problems, are akin to tip-toeing through a marathon. Linear multistep methods were developed to accommodate the need for fewer function evaluations.
Variation: Numerical Integration
Published in Markus W. Covert, Fundamentals of Systems Biology, 2017
The midpoint method uses the Euler method to estimate the variable value at tm (halfway between t0 and t1) and to determine the instantaneous rate of change at that value to generate a slope. The midpoint method is called second order because its error is estimated to be proportional to the cube of the time step.
Random field modelling of spatial variability in concrete – a review
Published in Structure and Infrastructure Engineering, 2023
Wouter Botte, Eline Vereecken, Robby Caspeele
In (Stewart & Suo, 2009), element lengths are chosen to be dependent on: (i) the ability of corroded reinforcement to redistribute stresses to adjacent reinforcement via the concrete matrix, (ii) mechanical behaviour of the reinforcement, (iii) anchorage length of reinforcement, and (iv) geometry and spacing of reinforcement, resulting in element lengths between 100 and 1000 mm. The applied discretization method was the midpoint method. For a 1D beam with a shear failure mode influenced by pitting corrosion, an element length equal to the effective depth of the beam is suggested. For the investigated case, this corresponded to an element length of 750 mm, for a scale of fluctuation equal to 2 m and a Gaussian correlation model.