Euler backward

For the spatial solution of the nonlinear coupled multi-field problem given in Sect 1.3, the Finite Element Method (FEM) is applied. The equations for the three fields are solved with a Newton-Raphson algorithm, and the time integration is performed with the implicit Euler backwards scheme. 

The Euler backward method, for example, is A-stable since... 

It seems that the trapezium algorithm is superior to the Euler backward method from all viewpoints Even though they are both A-stable and need to solve a nonlinear system in y +i, the trapezium algorithm is of second order, while Euler backward is of first order only. However, there is also an additional aspect to consider. 

For example, the Euler backward is strongly A-stable, whereas the trapeziimi algorithm is not. Actually, the trapezium algorithm has x - 1, while hX — oo. ... 

This technique is called upwind and takes its name from fluid-dynamic applications. Actually, in these problems the direction of the stable integration corresponds to moving against the wind (Ascher and Petzold, 1998). The selection can also be seen as the application of an Euler backward method for the well-conditioned components for increasing and Euler forward for the stable components with decreasing x. 

To summarize the comparison of methods, we Ulustrate in Fig. 7.5 the percentage relative error between the numerical solutions and the exact solution for a fixed step size of 0.5. With this step size all three methods, explicit Euler, backward Euler, and trapezoid, produced integration stabiUty, but the relative error is certainly unacceptable. 

Thus, Eq. 12.131, Eq. 12.130 for i = 2,3,..., IV — 2, and Eq. 12.132 will form a set of N - 1 equations with AT - 1 unknowns (y, y2, , yv- )- This set of coupled ordinary differential equations can be solved by any of the integration solvers described in Chapter 7. Alternately, we can apply the same finite difference procedure to the time domain, for a completely iterative solution. This is essentially the Euler, backward Euler and the Trapezoidal rule discussed in Chapter 7, as we shall see. 

For example, the SHAKE algorithm [17] freezes out particular motions, such as bond stretching, using holonomic constraints. One of the differences between SHAKE and the present approach is that in SHAKE we have to know in advance the identity of the fast modes. No such restriction is imposed in the present investigation. Another related algorithm is the Backward Euler approach [18], in which a Langevin equation is solved and the slow modes are constantly cooled down. However, the Backward Euler scheme employs an initial value solver of the differential equation and therefore the increase in step size is limited. 

Implicit Methods By using different interpolation formulas involving y, it is possible to cferive imphcit integration methods. Implicit methods result in a nonhnear equation to be solved for y so that iterative methods must be used. The backward Euler method is a first-order method. 

To solve the general problem using the backward Euler method, replace the nonlinear differential equation with the nonhuear algebraic equation for one step. 

A major complication exists for constructing the Lagrangian density of a pair of particles diffusing relative to each other. The diffusion (Euler) equation is dissipative and the density of the diffusing species is not conserved. The Euler density, p, would lead to a space—time invariant, Sfr, which would not be constant. This difficulty requires the same approach as that used to handle the Schrodinger equation. Morse and Feshbach [499] define a reverse or backward diffusion equation where time goes backwards compared with that in eqn. (254)... 

This approach to defining the Lagrangian density with the aid of both forward and backward Euler densities ip and ip uses the neat construct that ip ip is time-invariant. This is as true in the quantum mechanical analogy. 

Implicit methods, which have far better stability properties than explicit methods, provide the computational approach to solving stiff problems. The simplest implicit method is the backward (implicit) Euler method, which is stated as... 

As discussed in Section 15.3.2 on the implicit solution of transient differential equations, one step of the backward Euler method takes the form... 

C. S. Peskin and T. Schlick, Comm. Pure Appl. Math., 42,1001 (1989). Molecular Dynamics by the Backward-Euler Method. 

Equation (92) is simpler than (60) but otherwise structurally identical to it. In particular, the Euler-Lagrange equation associated with minimizing (92) has the form of a backward Kolmogorov equation ... 

Backward differences, first order implicit Euler method ... 

If the integral on the right side of (12.55) is estimated using the value of the integrand at the final point, we obtain the first order implicit or backward Euler Method ... 

Implicit methods have been developed to overcome this problem. The backward Euler method is... 

Adams-Moulton Family of Methods One-step Adams-Moulton, backward Euler s rule... 

However, the implicit, backward Euler approximation gives... 

This algorithm is the backward Euler ox fully implicit Euler method. Because this formula is implicit, a system of algebraic equations must be solved to calculate c"+l from cf for i — 1,2,..., N. This step is expensive because the Jacobian of the ODE system needs to be inverted, a process that requires on the order of N3 operations. This inversion should in principle be repeated in each step. 

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