Question

Stability of Numerical Methods Numerical methods are often tested on simple initial value problems of the form

y'+λy=0,y(0)=1,(λ= constant ), which has the solution ϕ(x)=e −λˣNotice that for each λ>0 the solution ϕ(x) tends to zero as x→+[infinity]. Thus, a desirable property for any numerical scheme that generates approximations
Notice that for each λ > 0, the solution ϕ(x) tends to zero as x → +[infinity]. Thus, a desirable property for any numerical scheme that generates approximations y₀, y₁, y₂, y₃, … to ϕ(x) at the points 0, h, 2h, 3h, … is that, for λ > 0,

Answer 1

The stability of numerical methods refers to their ability to generate accurate approximations to the solution of a differential equation over a long period of time.

Step-by-step explanation:

The stability of numerical methods refers to their ability to generate accurate approximations to the solution of a differential equation over a long period of time. For the initial value problem y' + λy = 0, y(0) = 1, where λ is a constant, the solution ϕ(x) = e-λx tends to zero as x approaches positive infinity. A desirable property for a numerical scheme is that the generated approximations y₀, y₁, y₂, y₃, ... to ϕ(x) at the points 0, h, 2h, 3h, ... remain stable for λ > 0.