In this post I wanted to document the behavior of several different numerical methods using a set of benchmark test problems.

### Test Problems

• Ball sliding on plane
• Block sliding on plane
• Block sliding from rest on incline $65^\circ$
• Block coming to resting on incline $25^\circ$
• Ball falling on stack of 20 balls

### Solvers

Solver (Matlab)   Solver (Gams)
Nesterov   Conopt
Barzilai-Borwein   IPopt
Jacobi   Path

## Test Problems

### Ball sliding on plane

In this test problem there is a ball in perfect contact with a plane, the ball has an initial velocity of $-2 \frac{m}{s}$ in the $x$ direction. friction is .1, step size = .005, with a step size of .01, path returns a weird solution.

The next plots shows the rotational velocity in the z direction

### Box sliding on plane

In this test problem there is a box in perfect contact with a plane, the box has an initial velocity of $-2 \frac{m}{s}$ in the $x$ direction. The friction coefficient is .3;s

### Box sliding on $65^\circ$ Incline

In this test problem there is a box in perfect contact with a plane, it starts at rests and then slides down for one second. friction is .2

### Box sliding to a stop on $25^\circ$ Incline

In this test problem there is a box in perfect contact with a plane, it starts with velocity of $-2 \frac{m}{s}$ parallel to the incline. It sides and comes to a stop, friction is .6

### Stack of 20 balls with heavy mass

A stack of 20 spheres with a larger heavier sphere dropping on top

Velocity of the top ball: