Linear Multistep Methods

Linear Multistep Methods With these methods the following notation is used:

Rather than using a single old value yn, additional old y values are used. Here are three examples all of which are widely used:

The first two are implicit methods because the right hand side contains ƒn+1 and hence yn+1. The third method is explicit. All three use two previous values in calculating yn+1 i.e. yn and yn-1 and are called 2-step methods. The names of the methods are

• the2-step backward differentiation formula 2-step backward differentiation formula used in the famous Gear’s method,

• the 2-step Adams Moulton method 2-step Adams Moulton method used in Adams code,

• the 2-step Adams Bashforth method 2-step Adams Bashforth method sometimes used as a real time integrator.

Simcenter Amesim methods vary a great deal in their characteristics. For most problems there will be some category of Simcenter Amesim method which is suitable.

They are much more difficult to implement as computer code than ERK methods. The code is extremely lengthy but the methods do show the following advantages over ERK methods:

• It is much easier to switch between Simcenter Amesim methods of different orders than with ERK methods. All the widely used implementations of Simcenter Amesim methods are variable step and variable order algorithms.

• There are well-developed Simcenter Amesim methods which can solve stiff problems efficiently. No ERK method can solve such problems efficiently.

There are three Simcenter Amesim algorithms which are widely used:

• Adams code Adams code is an excellent algorithm for non-stiff problems. It uses Adams-Moulton methods with orders up to 12 [3]. This gives great flexibility for low and high accuracy solutions.

• Gear’s method Gear’s method is the probably the best algorithm for stiff problems [1]. It uses methods known as backward differentiation formulae of orders 1 to 5.

• LSODA LSODA aims to get the best features of Adams code and Gears method in a single algorithm [4]. Integration starts with the Adams code methods. The characteristics of the equations are monitored and if stiffness is detected, a switch is made to Gear’s method. Monitoring of the characteristics of the equations continues switching between Adams-Moulton methods and backward differentiation formulae as the characteristics of the equations change.

The main disadvantage of Simcenter Amesim methods is that they are relatively intolerant of discontinuities.