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Exercise 1 (Consistency and autonomization of explicit and implicit RK methods; 6 points) Consider a Runge-Kutta method defined by the triple (B,c,a), given in the Butcher table with s stages and assume that B is non-singular. a) Show that an explicit Runge-Kutta scheme is consistent for all f E C(,Rd) if and only if b) Prove that an explicit Runge-Kutta method is invariant under autonomization if and only if it is consistent and satisfies i-1 <=p >bii, i=1 ..S j=1 c) Now consider an implicit Runge-Kutta scheme. Show that the coefficients of an implicit Runge-Kutta scheme defined by collocation satisfy (with the convention 0=1) (i) D5=1 a-1cj=k,k=1,.,s. .,s In particular, the method is consistent and invariant under autonomization.

Name: exercise 1 consistency and autonomization of explicit and implicit rk methods 6 points consider a runge kutta method defined by the triple bca given in the butcher table with s stages and as 45255
Uploaded: 2023-02-28T10:49:13-08:00
Duration: 1 min 16 s
Channel: Sri K
Description: exercise 1 consistency and autonomization of explicit and implicit rk methods 6 points consider a runge kutta method defined by the triple bca given in the butcher table with s stages and as 45255

          Exercise 1 (Consistency and autonomization of explicit and implicit RK methods; 6 points)
Consider a Runge-Kutta method defined by the triple (B,c,a), given in the Butcher table with s stages and assume that B is non-singular.
a) Show that an explicit Runge-Kutta scheme is consistent for all f E C(,Rd) if and only if
b) Prove that an explicit Runge-Kutta method is invariant under autonomization if and only if it is consistent and satisfies
i-1
<=p >bii, i=1 ..S j=1
c) Now consider an implicit Runge-Kutta scheme. Show that the coefficients of an implicit Runge-Kutta scheme defined by collocation satisfy (with the convention 0=1)
(i) D5=1 a-1cj=k,k=1,.,s.
.,s
In particular, the method is consistent and invariant under autonomization.

exercise 1 consistency and autonomization of explicit and implicit rk methods 6 points consider a runge kutta method defined by the triple bca given in the butcher table with s stages and as 45255

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