Algorithms backward Euler

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. 

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. 

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 ... 

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. 

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. 

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. ... 

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