Solver for scientific computing that detects and handles discontinuities or irregularities in a simulation
Methods, systems, and computing devices of a solver for detecting discontinuities or irregularities in a simulation in scientific computing is disclosed. The solver detects a discontinuity or irregularity using two phases: a crude phase, and a fine phase. The crude phase determines an estimate of where the discontinuity is located by extrapolating the interpolant. The fine phase iteratively refines the location of the discontinuity using root-finding and simplified Newton iterations. The solver disclosed herein takes fewer steps when running a simulation because it accurately steps over the discontinuities.
1. A method for detecting discontinuities in a simulation by a solver in scientific computing, the method comprising:
constructing a collocation polynomial over an integration interval;
determining an estimate of a point within the integration interval where a discontinuity is located; and
refining the estimate of the point where the discontinuity is located, wherein the estimate is refined using root-finding and simplified Newton iterations.
2. The method of claim 1 , wherein the collocation polynomial represents a non-linear system being modeled by the simulation.
3. The method of claim 1 , wherein the estimate is determined based on where a solution to the collocation polynomial equals zero.
4. The method of claim 1 , wherein a root-finding problem for the root-finding is generated from a symbolic domain representation, wherein the symbolic domain representation is scanned for irregularities.
5. The method of claim 1 , wherein step size is placed such that the discontinuity is captured by a step of the solver.
6. The method of claim 1 , wherein the root-finding is performed on the collocation polynomial to refine an initial guess of an estimate of a solution of the collocation polynomial.
7. A method for detecting discontinuities in a simulation by a solver in scientific computing, the method comprising:
constructing a collocation polynomial over an integration interval;
determining an estimate of a point within the integration interval where a discontinuity is located; and
refining the estimate of the point where the discontinuity is located, wherein the estimate is refined using root-finding and simplified Newton iterations, and wherein refining the estimate of the point where the discontinuity is located comprises:
performing simplified Newton iterations with an estimate of a step size;
performing root-finding via the collocation polynomial to update the estimate of the step size; and
while the step size has not converged:
performing additional simplified Newton iterations using the updated estimate of the step size; and
performing additional root-finding via the collocation polynomial to update the estimate of the step size.
8. The method of claim 7 , wherein the collocation polynomial represents a non-linear system being modeled by the simulation.
9. The method of claim 7 , wherein the estimate is determined based on where a solution to the collocation polynomial equals zero.
10. The method of claim 7 , wherein a root-finding problem for the root-finding is generated from a symbolic domain representation, wherein the symbolic domain representation is scanned for irregularities.
11. The method of claim 7 , wherein step size is placed such that the discontinuity is captured by a step of the solver.
12. The method of claim 7 , wherein an initial value for the estimate of the step size is determined by extrapolating an interpolant of the collocation polynomial.
13. The method of claim 7 , wherein the point within the integration interval where the discontinuity is located is found when step size converges.
14. A computing device with a solver for detecting discontinuities in a simulation by in scientific computing, the computing device comprising:
a memory; and
at least one processor configured for:
constructing a collocation polynomial over an integration interval;
determining an estimate of a point within the integration interval where a discontinuity is located; and
refining the estimate of the point where the discontinuity is located, wherein the estimate is refined using root-finding and simplified Newton iterations, and wherein refining the estimate of the point where the discontinuity is located comprises:
performing simplified Newton iterations with an estimate of a step size;
performing root-finding via the collocation polynomial to update the estimate of the step size; and
while the step size has not converged:
performing additional simplified Newton iterations using the updated estimate of the step size; and
performing additional root-finding via the collocation polynomial to update the estimate of the step size.
15. The computing device of claim 14 , wherein the collocation polynomial represents a non-linear system being modeled by the simulation.
16. The computing device of claim 14 , wherein the estimate is determined based on where a solution to the collocation polynomial equals zero.
17. The computing device of claim 14 , wherein a root-finding problem for the root-finding is generated from a symbolic domain representation, wherein the symbolic domain representation is scanned for irregularities.
18. The computing device of claim 14 , wherein step size is placed such that the discontinuity is captured by a step of the solver.
19. The computing device of claim 14 , wherein the root-finding is performed on the collocation polynomial to refine an initial guess of the estimate of the solution of the collocation polynomial.
20. The computing device of claim 14 , wherein the simplified Newton iterations are performed by an implicit stiff solver.