Explicit Numerical Integration Algorithms

EULER ALGORITHM. The simplest possible numerical-integration scheme (and the most useful) is Euler (pronounced oiler ), illustrated in Fig. 4.7. Assume we wish to solve the ODE 

If the step size (the integration interval) is small enough, this estimate of x will be very close to the correct value. 

Euler integration is extremely simple to program, as will be illustrated in Example 4,3. This simplicity is retained, even as the number of ODEs increases and as the derivative functions become more complex and nonlinear. 

If we have two simultaneous, coupled ODEs to numerically integrate 

Notice that only one derivative evaluation is required per ODE at each point in time. If we had a set of N ordinary differential equations, we would have N equations like Eqs. (4.50) and (4.51). 

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As discussed in the introduction to this chapter, the solution of ordinary differential equations (ODEs) on a digital computer involves numerical integration. We will present several of the simplest and most popular numerical-integration algorithms. In Sec, 4.4.1 we will discuss explicit methods and in Sec. 4.4.2 we will briefly describe implicit algorithms. The differences between the two types and their advantages and disadvantages will be discussed. 

A large number of explicit numerical advection algorithms were described and evaluated for the use in atmospheric transport and chemistry models by Rood [162], and Dabdub and Seinfeld [32]. A requirement in air pollution simulations is to calculate the transport of pollutants in a strictly conservative manner. For this purpose, the flux integral method has been a popular procedure for constructing an explicit single step forward in time conservative control volume update of the unsteady multidimensional convection-diffusion equation. The second order moments (SOM) [164, 148], Bott [14, 15], and UTOPIA (Uniformly Third-Order Polynomial Interpolation Algorithm) [112] schemes are all derived based on the flux integral concept. 

The explicit expression for defined in equation (9.21) may be derived by requiring that for all k the vanishes at all times. Using a certain numerical integration algorithm, the set of Lagrangian multipliers is no longer evaluated by equation (9.21) but by equations that contain free parameters corresponding to the. These... 

Another difference is related to the mathematical formulation. Equation (1) is deterministic and does not include explicit stochasticity. In contrast, the equations of motion for a Brownian particle include noise. Nevertheless, similar algorithms are adopted to solve the two differential equations as outlined below. The most common approach is to numerically integrate the above differential equations using small time steps and preset initial values. 

To perform high-dimensional integrals numerically, the Monte Carlo (MC) method is often used. The MC method turns a multidimensional integral into a sum over a stochastic sequence of points called a trajectory, so that in the limit of an infinitely long trajectory, the value of the integral is numerically reproduced. Explicitly, the MC algorithm yields the ratio of two integrals... 

The fundamental difficulty in solving DEs explicitly via finite formulas is tied to the fact that antiderivatives are known for only very few functions / M —> M. One can always differentiate (via the product, quotient, or chain rule) an explicitly given function f(x) quite easily, but finding an antiderivative function F with F x) = f(x) is impossible for all except very few functions /. Numerical approximations of antiderivatives can, however, be found in the form of a table of values (rather than a functional expression) numerically by a multitude of integration methods such as collected in the ode... m file suite inside MATLAB. Some of these numerical methods have been used for several centuries, while the algorithms for stiff DEs are just a few decades old. These codes are... 

The change in velocity vt is equal to the integral of acceleration over time. In MolD, one numerically and iteratively integrates the classical equations of motion for every explicit atom N in the system by marching forward in time by tiny time increments, At. A number of algorithms exist for this purpose (Brooks et al., 1988 McCammon and Harvey, 1987), and the simplest formulation is shown below ... 

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