Impact of a discontinuity on simulation

Simcenter Amesim solvers are variable step solvers. These solvers are based on linear multi-step methods ([1] and [2]) and require the state equations be at least continuous, and continuously differentiable.

Any kind of discontinuity affecting a state equation may imply a sharp change in the derivative of the state variable. A variable step solver will of course be able to handle this sharp change. But this will imply a strong reduction of the integration step size, leading to long simulation time.

Figure 2: continuous curve with discontinuous derivative

The characteristics of the variable are based on 2 distinct curves C1 and C2. The variable is continuous but its derivative is discontinuous at point P. Integration along C1 or C2 is fine, but problems arise when reaching P.

Using a simple algorithm as in the following pseudo code

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If (x < x*)
   Use formula for C1
Else
   Use formula for C2

will force the integrator to drastically reduce its step near the point P in a desperate attempt to meet accuracy requirements. Many integrators will give up with an error message when the step gets too small. Others will have a minimum step size and when this is reached will carry on, even though the accuracy requirements are not met. The first approach leads to frustration and the second can lead to highly inaccurate results.

The principle of discontinuity handling is simple in concept. If the solution is on curve C1 and goes past the point P then the equation for C2 is not used. Instead the equation for C1 is still used so that effectively the curve C1’ is employed. This gives rise to a point Q being predicted. After convergence is obtained, the integrator must realize that a discontinuity point has been passed over and start trying to locate it.

Figure 3: C1 curve continuous extrapolation

A specific communication mechanism is then introduced between submodels and solver. This mechanism allows the solver to locate precisely (to machine accuracy) the simulation time at which the discontinuity occurs. The integration step is then adapted to a time just before this discontinuity.

After this the integrator restarts itself, forgetting all information to the left of point P, using the smooth curve C2.