Euler Backward Method

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. 

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. 

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... 

The forward Euler method is thus referred to as an explicit method, because Xk+i is taken to depend only on points that have been previously calculated, that is, x + depends only on The discretization used in the backward Euler method leads to the following approximation of the ODE... 

Because Xk+i appears on both sides of this equation, additional steps are required to solve for x +i before the approximation can be used to calculate it. (This can be done via iteration, such as through the Newton-Raphson method.) Hence, the backward Euler method is also referred to as an implicit method. The trapezoidal algorithm averages the information from the forward and backward Euler algorithms such that the iteration equation to be used is... 

Other examples of implicit Runge-Kutta algorithms include the backward Euler methods ... 

In dubious cases, restart with the backward Euler method, which is strongly A-stable. 

When the step must be suddenly reduced to very small values during integration, it is usefiil to reinitialize the problem. By doing so, we enjoy the following advantages The backward Euler method, strongly A-stable, is used and the memory of the step that generates the numerical disturbance is lost as well. 

If the problem is stiff, it is necessary to use an A-stable or, better still, strongly A-stable algorithm such as the backward Euler method (2.12) or the Cash method. 

As an alternative, consider the Backward Euler method q + = q + hp +i, p +i = p — hq +i. Then it is easily shown that both eigenvalues of the matrix lie in the interior of the unit disk, and hence all solutions of the recurrence relation tend to the origin with increasing n (the origin is an attractive equilibrium point). In this case, all densities evolve toward the Dirac distribution centered at the origin (5[ ](5 p]. The only (distributional) solution of Cjp = 0 is again 3[( ]5[p], which in this case is attractive. 

The backward Euler method yields a simpler implicit formula, and its recursive relation to calculate y is given as... 

Knowing the solution for yj at tj = 0.2, we can proceed in a similar fashion to obtain y>2, y, and so on. Figure 7.3 illustrates computations for a number of step sizes. No instability is observed for the case of backward Euler method. 

Figure 73 Plots of the numerical solutions for the backward Euler method.

To resolve the problem of stability, we approach the problem in the same way we did in the backward Euler method. We evaluate Eq. 12.130 at the unknown time level fy+j and use the following backward difference formula for the time derivative term (which is first order correct)... 

All the methods presented so far, e.g. the Euler and the Runge-Kutta methods, are examples of explicit methods, as the numerical solution aty +i has an explicit formula. Explicit methods, however, have problems with stability, and there are certain stability constraints that prevent the explicit methods from taking very large time steps. Stability analysis can be used to show that the explicit Euler method is conditionally stable, i.e. the step size has to be chosen sufficiently small to ensure stability. This conditional stability, i.e. the existence of a critical step size beyond which numerical instabilities manifest, is typical for all explicit methods. In contrast, the implicit methods have much better stability properties. Let us introduce the implicit backward Euler method. 

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