Week 6 - convergence, consistency & stability Flashcards
Q: Convergence of solution?
Week 6 - convergence, consistency & stability
Q: Convergence of solution?
the solution of Algebraic equations, which approximate a given PDE, is said to be convergent if the approximate solution approaches the exact solution of the PDE, for each value of independent variables (as grid spacing) tends to zero.
Q: Lax Equivalence theorem?
Q: Lax Equivalence theorem?
for a well-posed linear IVP & FD approximation that satisfies the consistency condition, stability is necessary & sufficient condition for convergence.
Q: Consistency? consistent System of equation?
Q: Consistency? consistent System of equation?
algebraic equation (obtained from discretization) is said to be consistent with the original PDE, if the limit that grid spacing tends to zero, system of algebraic equations is equivalent to PDE. necessary but not sufficient condition
Q: How to Test consistency?
Q: How to Test consistency?
on substituting exact soln to algebraic equations & after rearranging, it should give original PDE plus error term, which should reduce to zero as grid spacing reduces to zero.
Q: Stability?
Q: Stability?
Round off (because of finite number of significant figures) should not accumulate as we move forward in time. round off error at (n+1)th level can be written in terms of error at nth level.
Q: Von-Neuman Stability Analysis?
Lec-18
Q: Von-Neuman Stability Analysis?
most commonly used method of determining stability criterion.
gives a necessary not sufficient condition
main idea is to study the growth of perturbations/ waves with time, for a discretization scheme
put phi = exp{ i k x_(j) } & find | G |
1D Transient Heat Conduction FTCS - conditionally stable CN - unconditionally stable 2D Transient Heat conduction FTCS - conditionally stable Linear Advection Equation FTCS - unconditionally unstable FTBS - conditionally stable (assuming u>0) FTFS - conditionally stable (assuming u<0)
Q: CFL? Physical meaning of CFL criteria?
Q: CFL? Physical meaning of CFL criteria?
