Newton’s Laws of Motion:
- 1st Law: In the absence of a net force, a body either is at rest or moves in a straight line with constant speed.
- 2nd Law: A body experiencing a force F experiences an acceleration a related to F by F = ma, where m is the mass of the body. Alternatively, force is equal to the time derivative of momentum.
- 3rd Law: Whenever a first body exerts a force F on a second body, the second body exerts a force -F on the first body. F and -F are equal in magnitude and opposite in direction.
Newton’s World is based on the following incremental equations of motion with smallest unit of time :
as another way of writing
which combined with Newton’s 2nd Law assuming , take the form:
These equations can be solved by time-stepping with time step :
and , and are position, velocity and acceleration at time after successive time steps with time step . By time-stepping forward in time and at time are computed from and already computed at the preceding time .
Euler’s Method: Forward Euler
With each tick of time, velocity and position are thus updated according to
from given initial values and at initial time , where is the force acting on the body at time . We refer to this update formula as Forward Euler.
An alternative update formula is obtained by updating first velocity to and using this value when updating to :
which we will refer to as Smart-Euler. You will soon discover the difference between Euler and Smart-Euler.
A variant of Smart-Euler is
where the mean velocity is used instead of either or .
Below we shall meet variants with depending on . The basic method of this form is the Trapezoidal Method:
where and , which requires iteration because depends on ,which depends on .
We compare with Backward Euler:
which also requires iteration.
A variant of the Trapezoidal Method is Midpoint Euler:
with evaluated at the midpoint between and , also requiring iteration.
Explicit vs Implicit Methods
We distinguish between explicit methods like Forward Euler and Smart Euler with direct update, and implicit methods requiring iteration, like the Trapezoidal Method, Backward and Midpoint Euler, where the update formula for and is repeated with latest values inserted in the right hand side.
With a (small) fixed number of iterations, implicit methods can be viewed as explicit direct update methods.
First and Second Order Accurate Methods
Forward and Backward Euler are first order accurate in the sense that the time-stepping error is proportional to the time step , while the Trapezoidal Method and Midpoint Euler are second order accurate in the sense that the time stepping error is proportional to the time-step squared , which is much smaller than since is small. Smart Euler is rather second than first order accurate.