Work Sheet Assignment 13: Under the conditions (i) and (ii) of THM 3.6, prove (global) convergence order p = 1 of outer Theta methods (OTM) on ID = IRd , governed by the scheme yn+1 = yn + hn h θf(tn+1, yn+1) + (1 − θ)f(tn, yn) i applied to IVPs dx/dt = f(t, x), x(0) = x0 = y0 with f ∈ C 0 Lip(L) ([0, T] × IRd ) along any partitions (tn) n∈IN of [0, T] while θ ∈ [0, 1] is any real constant, hn < 1 for all n ∈ IN (Take V (x) = 1 + kxkd). Compute the leading (global) error constant Kg^c of Theta methods. under the condition θhn < 1 to avoid numerical ”explosions”. Hint: You should apply FTNA (Thm 3.5) and check the fulfillment of all axioms (i)-(v) while ID = IRd is presumed (Note that you need to check V -stability and local consistency order p = 2, and use Lemma 3.3). Provide all mathematical details step by step (not just final answers and not just verbal arguments mostly).
Work Sheet Assignment 13: Under the conditions (i) and (ii) of THM 3.6, prove (global) convergence order p = 1 of outer Theta methods (OTM) on ID = IRd , governed by the scheme yn+1 = yn + hn h θf(tn+1, yn+1) + (1 − θ)f(tn, yn) i applied to IVPs dx/dt = f(t, x), x(0) = x0 = y0 with f ∈ C 0 Lip(L) ([0, T] × IRd ) along any partitions (tn) n∈IN of [0, T] while θ ∈ [0, 1] is any real constant, hn < 1 for all n ∈ IN (Take V (x) = 1 + kxkd). Compute the leading (global) error constant Kg^c of Theta methods. under the condition θhn < 1 to avoid numerical ”explosions”. Hint: You should apply FTNA (Thm 3.5) and check the fulfillment of all axioms (i)-(v) while ID = IRd is presumed (Note that you need to check V -stability and local consistency order p = 2, and use Lemma 3.3). Provide all mathematical details step by step (not just final answers and not just verbal arguments mostly).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Work Sheet Assignment 13: Under the conditions (i) and (ii) of THM 3.6, prove (global)
convergence order p = 1 of outer Theta methods (OTM) on ID = IRd
, governed by the
scheme
yn+1 = yn + hn
h
θf(tn+1, yn+1) + (1 − θ)f(tn, yn)
i
applied to IVPs dx/dt = f(t, x), x(0) = x0 = y0 with f ∈ C
0
Lip(L)
([0, T] × IRd
) along any
partitions (tn)
n∈IN of [0, T] while θ ∈ [0, 1] is any real constant, hn < 1 for all n ∈ IN (Take
V (x) = 1 + kxkd). Compute the leading (global) error constant Kg^c
of Theta methods. under
the condition θhn < 1 to avoid numerical ”explosions”. Hint: You should apply FTNA (Thm
3.5) and check the fulfillment of all axioms (i)-(v) while ID = IRd
is presumed (Note that you
need to check V -stability and local consistency order p = 2, and use Lemma 3.3). Provide
all mathematical details step by step (not just final answers and not just verbal arguments
mostly).
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