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❆❞❛♣t✐✈❡ ❚✐♠❡ ❉✐s❝r❡t✐③❛t✐♦♥ ❢♦r ❘❡t❛r❞❡❞ P♦t❡♥t✐❛❧s

❙t❡❢❛♥ ❙❛✉t❡r✱ ❯♥✐✈❡rs✐tät ❩ür✐❝❤

❥♦✐♥t ✇♦r❦ ✇✐t❤ ❆❧❡①❛♥❞❡r ❱❡✐t✱ ❯♥✐✈❡rs✐t② ♦❢ ❈❤✐❝❛❣♦

❘■❈❆▼✱ ▲✐♥③ ◆♦✈❡♠❜❡r ✶✵✱ ✷✵✶✻

❖✉t❧✐♥❡

■♥t❡❣r❛❧ ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ✇❛✈❡ ❡q✉❛t✐♦♥ ❉✐s❝r❡t✐③❛t✐♦♥ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❆ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t✐♦♥ ✐♥ t✐♠❡ ◆✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ❢♦r t❤❡ ❛❞❛♣t✐✈❡ ❛❧❣♦r✐t❤♠

■♥tr♦❞✉❝t✐♦♥

ui ue ue ue Ωi Γ Ω n

❋✐❣✉r❡✿ ❙❝❛tt❡r✐♥❣ ♣r♦❜❧❡♠ ✐♥ ✉♥❜♦✉♥❞❡❞ ❡①t❡r✐♦r ❞♦♠❛✐♥s✳

■♥tr♦❞✉❝t✐♦♥

▲❡t Ω ⊂ R✸ ❜❡ ❛ ▲✐♣s❝❤✐t③ ❞♦♠❛✐♥ ✇✐t❤ ❜♦✉♥❞❛r② Γ✳ ■♥ s❝❛tt❡r✐♥❣ ♣r♦❜❧❡♠s Ω ✐s t②♣✐❝❛❧❧② ❛♥ ✉♥❜♦✉♥❞❡❞ ❞♦♠❛✐♥✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❤♦♠♦❣❡♥❡♦✉s ✇❛✈❡ ❡q✉❛t✐♦♥ ∂✷

t u − ∆u = ✵

✐♥ Ω × [✵, T] ✇✐t❤ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s u(·, ✵) = ∂tu(·, ✵) = ✵ ✐♥ Ω ❛♥❞ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s u = g ♦♥ Γ × [✵, T] ♦♥ ❛ t✐♠❡ ✐♥t❡r✈❛❧ [✵, T] ❢♦r s♦♠❡ T > ✵✳

■♥t❡❣r❛❧ ❋♦r♠✉❧❛t✐♦♥

❲❡ ❡♠♣❧♦② ❛♥ ❛♥s❛t③ ❛s ❛ s✐♥❣❧❡ ❧❛②❡r ♣♦t❡♥t✐❛❧ u(x, t) : = (Sφ)(x, t) = t

✵

• Γ

δ(t − τ − x − y) ✹πx − y φ(y, τ) dΓydτ =

• Γ

φ(y, t − x − y) ✹πx − y dΓy, (x, t) ∈ Ω × [✵, T]. ❚❤❡ ❡①t❡♥s✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s✳ ❚❤❡r❡❢♦r❡ t❤❡ ✉♥❦♥♦✇♥ ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✐s ❞❡t❡r♠✐♥❡❞ ✈✐❛ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❚❤✐s ❧❡❛❞s t♦ t❤❡ ❜♦✉♥❞❛r② ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ✹ ♦♥ ✵

■♥t❡❣r❛❧ ❋♦r♠✉❧❛t✐♦♥

❲❡ ❡♠♣❧♦② ❛♥ ❛♥s❛t③ ❛s ❛ s✐♥❣❧❡ ❧❛②❡r ♣♦t❡♥t✐❛❧ u(x, t) : = (Sφ)(x, t) = t

✵

• Γ

δ(t − τ − x − y) ✹πx − y φ(y, τ) dΓydτ =

• Γ

φ(y, t − x − y) ✹πx − y dΓy, (x, t) ∈ Ω × [✵, T]. ❚❤❡ ❡①t❡♥s✐♦♥ x → Γ ✐s ❝♦♥t✐♥✉♦✉s✳ ❚❤❡r❡❢♦r❡ t❤❡ ✉♥❦♥♦✇♥ ❞❡♥s✐t② ❢✉♥❝t✐♦♥ φ ✐s ❞❡t❡r♠✐♥❡❞ ✈✐❛ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❚❤✐s ❧❡❛❞s t♦ t❤❡ ❜♦✉♥❞❛r② ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥

• Γ

φ(y, t − x − y) ✹πx − y dΓy = g(x, t) ♦♥ Γ × [✵, T].

❱❛r✐❛t✐♦♥❛❧ ❋♦r♠✉❧❛t✐♦♥ ❛♥❞ ❉✐s❝r❡t✐③❛t✐♦♥

❆ ❝♦❡r❝✐✈❡ s♣❛❝❡✲t✐♠❡ ✈❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥ ✐s ❣✐✈❡♥ ❜②✿ ❋✐♥❞ φ s✳t✳ T

✵

• Γ
• Γ

˙ φ(y, t − x − y)ζ(x, t) ✹πx − y dΓydΓxdt = T

✵

• Γ

˙ g(x, t)ζ(x, t)dΓxdt ❢♦r ❛❧❧ ζ✳ ❲❡ ✉s❡ ❛ ●❛❧❡r❦✐♥ ♠❡t❤♦❞ ✐♥ s♣❛❝❡ ❛♥❞ t✐♠❡ ✐♥ ♦r❞❡r t♦ ❞✐s❝r❡t✐③❡ t❤✐s ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥✳ ❚❤✉s ✇❡ ♠❛❦❡ t❤❡ ❞✐s❝r❡t❡ ❛♥s❛t③

• ❛❧❡r❦✐♥

✶ ✶

❱❛r✐❛t✐♦♥❛❧ ❋♦r♠✉❧❛t✐♦♥ ❛♥❞ ❉✐s❝r❡t✐③❛t✐♦♥

❆ ❝♦❡r❝✐✈❡ s♣❛❝❡✲t✐♠❡ ✈❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥ ✐s ❣✐✈❡♥ ❜②✿ ❋✐♥❞ φ s✳t✳ T

✵

• Γ
• Γ

˙ φ(y, t − x − y)ζ(x, t) ✹πx − y dΓydΓxdt = T

✵

• Γ

˙ g(x, t)ζ(x, t)dΓxdt ❢♦r ❛❧❧ ζ✳ ❲❡ ✉s❡ ❛ ●❛❧❡r❦✐♥ ♠❡t❤♦❞ ✐♥ s♣❛❝❡ ❛♥❞ t✐♠❡ ✐♥ ♦r❞❡r t♦ ❞✐s❝r❡t✐③❡ t❤✐s ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥✳ ❚❤✉s ✇❡ ♠❛❦❡ t❤❡ ❞✐s❝r❡t❡ ❛♥s❛t③ φ●❛❧❡r❦✐♥(x, t) =

N

• i=✶

M

• j=✶

αj

iϕj(x)bi(t).

■♥s❡rt✐♥❣ t❤✐s ❛♥s❛t③ ✐♥ t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ❧❡❛❞s t♦✿ ❋✐♥❞ αj

i, i = ✶, . . . , N, j = ✶, . . . , M s✉❝❤ t❤❛t N

• i=✶

M

• j=✶

αj

i

• Γ
• Γ

ϕj(y) ϕl(x) ✹πx − y ψk,i(x − y)dΓydΓx = T

✵

• Γ

˙ g(x, t) ϕl(x) bk(t)dΓxdt ❢♦r k = ✶, . . . , N ❛♥❞ l = ✶, . . . , M ❛♥❞ ψk,i(r) := T

✵

˙ bi(t − r)bk(t)dt ✇❤❡r❡ r ∈ R>✵✳ ❲❡ ❝♦♥str✉❝t s♠♦♦t❤ ❛♥❞ ❝♦♠♣❛❝t❧② s✉♣♣♦rt❡❞ ❜❛s✐s ❢✉♥❝t✐♦♥s ✐♥ t✐♠❡ ✉s✐♥❣ t❤❡ P❛rt✐t✐♦♥ ♦❢ ❯♥✐t② ▼❡t❤♦❞✳

■♥s❡rt✐♥❣ t❤✐s ❛♥s❛t③ ✐♥ t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ❧❡❛❞s t♦✿ ❋✐♥❞ αj

i, i = ✶, . . . , N, j = ✶, . . . , M s✉❝❤ t❤❛t N

• i=✶

M

• j=✶

αj

i

• Γ
• Γ

ϕj(y) ϕl(x) ✹πx − y ψk,i(x − y)dΓydΓx = T

✵

• Γ

˙ g(x, t) ϕl(x) bk(t)dΓxdt ❢♦r k = ✶, . . . , N ❛♥❞ l = ✶, . . . , M ❛♥❞ ψk,i(r) := T

✵

˙ bi(t − r)bk(t)dt ✇❤❡r❡ r ∈ R>✵✳ ❲❡ ❝♦♥str✉❝t s♠♦♦t❤ ❛♥❞ ❝♦♠♣❛❝t❧② s✉♣♣♦rt❡❞ ❜❛s✐s ❢✉♥❝t✐♦♥s ✐♥ t✐♠❡ ✉s✐♥❣ t❤❡ P❛rt✐t✐♦♥ ♦❢ ❯♥✐t② ▼❡t❤♦❞✳

❆ s♠♦♦t❤ ♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t②

✶ ✵ τ✶ τ✷ τ✸ τ✹ τ✺ τ✻ τ✼ Θ✶ Θ✷ Θ✸ Θ✹ Θ✺ Θ✻ Θ✼ Θ✽

❋✐❣✉r❡✿ P❛rt✐t✐♦♥ ♦❢ ✉♥✐t② {ϕi} ❢♦r ✽ t✐♠❡st❡♣s✳

❇❛s✐s ❢✉♥❝t✐♦♥s ✐♥ t✐♠❡

❲❡ ❝♦♥s✐❞❡r ❛ ✭❧♦❝❛❧❧②✮ q✉❛s✐✲✉♥✐❢♦r♠ t✐♠❡ ♠❡s❤ ✇✐t❤ t✐♠❡st❡♣s ti s✳t✳ ✵ = t✵ < t✶ < t✷ < . . . < tN−✷ < tN−✶ = T. ❲❡ ♦❜t❛✐♥ ❜❛s✐s ❢✉♥❝t✐♦♥s ✐♥ t✐♠❡ ❜② ♠✉❧t✐♣❧②✐♥❣ t❤❡ ♣❛rt✐t♦♥ ♦❢ ✉♥✐t② ❢✉♥❝t✐♦♥s ✇✐t❤ s✉✐t❛❜❧② s❝❛❧❡❞ ▲❡❣❡♥❞r❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ♠❛①✐♠❛❧ ❞❡❣r❡❡ p✳ ϕ✶(t) t✷Pm−✷ ✷ t✶ t − ✶

• , m = ✷, . . . , ♠❛①(✷, p),

ϕi(t)Pm

• ✷ t − ti−✷

ti − ti−✷ − ✶

• , m = ✵, . . . , p,

i = ✷, . . . , N − ✶, ϕN(t)Pm

• ✷

t − tN−✷ tN−✶ − tN−✷ − ✶

• , m = ✵, . . . , p.

❇❛s✐s ♦❢ t❤❡ P❯▼ s♣❛❝❡

0.4 0.8 1.2 1.6 2 −0.5 0.5 1 m=0 m=1 m=2 m=3

❋✐❣✉r❡✿ ❇❛s✐s ❢✉♥❝t✐♦♥s ❢♦r t✵ = ✵✱ t✶ = ✵.✽ ❛♥❞ t✷ = ✷✳

❉✐s❝r❡t❡ Pr♦❜❧❡♠

❘❡❝❛❧❧ t❤❛t ✇❡ ✇❛♥t t♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠✿ ❋✐♥❞ αj

i, i = ✶, . . . , N, j = ✶, . . . , M s✉❝❤ t❤❛t N

• i=✶

M

• j=✶

Ak,i

j,l αj i = gk l ,

∀✶ ≤ k ≤ N ∀✶ ≤ l ≤ M, ✇❤❡r❡ Ak,i

j,l :=

• Γ
• Γ

ϕj(y) ϕl(x) ✹πx − y ψk,i(x − y)dΓydΓx ❛♥❞ ψk,i(r) = T

✵

˙ bi(t − r)bk(t)dt, gk

l =

T

✵

• Γ

˙ g(x, t) ϕl(x) bk(t)dΓxdt.

❙♣❛rs✐t② P❛tt❡r♥ ♦❢ ❆

❋✐❣✉r❡✿ ❙♣❛rs✐t② ♣❛tt❡r♥ ♦❢ ❆ ❢♦r Γ = [✵, ✷] ✇✐t❤ M = ✽✵ ❛♥❞ T = ✸ ✇✐t❤ N = ✸✵✳

❊✣❝✐❡♥t ❊✈❛❧✉❛t✐♦♥ ♦❢ ψk,i

❚❤❡ ❞✐r❡❝t ❡✈❛❧✉❛t✐♦♥ ♦❢ ψk,i ✉s✐♥❣ ●❛✉ss✲▲❡❣❡♥❞r❡ q✉❛❞r❛t✉r❡ ✐s ❡①♣❡♥s✐✈❡✳ ❲❡ t❤❡r❡❢♦r❡ r❡♣r❡s❡♥t ψk,i ♦♥ ✐ts s✉♣♣♦rt ❜② ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧s✳ ▲❡t s✉♣♣ ψk,i = [a, b] ❛♥❞ ∆l,j := [a + (j − ✶)hl, a + jhl], hl = b − a l ❜❡ ❛ r❡❣✉❧❛r s✉❜❞✐✈✐s✐♦♥✳ ❚❤❡♥ ✇❡ ❛♣♣r♦①✐♠❛t❡ ψk,i(r)|∆l.j ≈

p−✶

• q=✵

cqTq(θ(r)) − ✶ ✷c✵, ✇❤❡r❡ Tq ❛r❡ ❈❤❡❜②s❤❡✈ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ q ❛♥❞ θ : ∆l,j → [−✶, ✶] ✐s ❛ s❝❛❧✐♥❣ ❢✉♥❝t✐♦♥✳

❊✣❝✐❡♥t ❊✈❛❧✉❛t✐♦♥ ♦❢ ψk,i

❲❡ ❝♦♥s✐❞❡r t❤❡ t✐♠❡ ❣r✐❞ t✵ = ✵, t✶ = ✷, t✷ = ✸, t✸ = ✹.✺, t✹ = ✼. ▲❡t ρt✵,t✶,t✷ ❛♥❞ ρt✷,t✸,t✹ ❜❡ ❜✉♠♣ ❢✉♥❝t✐♦♥s ❛s ❛❜♦✈❡ ❛♥❞ ❞❡♥♦t❡ ❜② b✶(t) := ρt✵,t✶,t✷(t), b✸(t) := ρt✵,t✶,t✷(t)P✸

• ✷ t − t✵

t✷ − t✵ − ✶

• ,

b✷(t) := ρt✷,t✸,t✹(t), b✹(t) := ρt✷,t✸,t✹(t)P✷

• ✷ t − t✷

t✹ − t✷ − ✶

• ,

❝♦rr❡s♣♦♥❞✐♥❣ ❜❛s✐s ❢✉♥❝t✐♦♥s✳ ❲❡ ❞❡✜♥❡ ψ✶(r) = ✼

✵

˙ b✶(t − r)b✷(t)dt ❛♥❞ ψ✷(r) = ✼

✵

˙ b✸(t − r)b✹(t)dt.

❊✣❝✐❡♥t ❊✈❛❧✉❛t✐♦♥ ♦❢ ψk,i

1 2 3 4 5 6 7 −1 −0.5 0.5 1 r ψ1(r) ψ2(r)

❋✐❣✉r❡✿ ψ✶(r) ❛♥❞ ψ✷(r)

20 40 60 80 100 10

−10

10

−8

10

−6

10

−4

10

−2

10 polynomial degree p Approximation error ψ1 Approximation error ψ2

❋✐❣✉r❡✿ ❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ✐♥ ❞❡♣❡♥❞❡♥❝❡ ♦❢ p ❢♦r l = ✶✳

◗✉❛❞r❛t✉r❡ ❡rr♦r ❛♥❛❧②s✐s

▲❡t I :=

• τ
• ˜

τ

ψ(x − y) ✹πx − y dΓydΓx ❛♥❞ Qn ❜❡ t❤❡ t❡♥s♦r ●❛✉ss✲▲❡❣❡♥❞r❡ q✉❛❞r❛t✉r❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ I ✇✐t❤ n q✉❛❞r❛t✉r❡ ♥♦❞❡s ✐♥ ❡❛❝❤ ❞✐r❡❝t✐♦♥✳ ■t ❤♦❧❞s✿

▲❡♠♠❛

▲❡t τ ❛♥❞ ˜ τ ❤❛✈❡ ♣♦s✐t✐✈❡ ❞✐st❛♥❝❡ ❛♥❞ γ ∈ (✵, ✷

✸)✳ ■❢ n ∈ N s❛t✐s✜❡s

❧♥(n)

✸ ✷ n− ✷ ✸+γ < C,

t❤❡♥ ✇❡ ❤❛✈❡ |I − Qn| ≤ ˜ C ·

• ❧♥(n)

❧♥(n) − ✷ · n−γ ❧♥(n)+✷, ✇❤❡r❡ C, ˜ C ❛r❡ ✭❡①♣❧✐❝✐t❧② ❦♥♦✇♥✮ ❝♦♥st❛♥ts✳

2 4 6 8 10 12 14 16 18 20 10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10 10

2

Number of quadrature nodes in each direction relative error Case 1 Case 2 Case 3 Case 4

❋✐❣✉r❡✿ ❈❛s❡ ✶✿ ■❞❡♥t✐❝❛❧ ♣❛♥❡❧s✱ ❈❛s❡ ✷✿ ❈♦♠♠♦♥ ❡❞❣❡✱ ❈❛s❡ ✸✿ ❘❡❣✉❧❛r ♥❡❛r ✜❡❧❞✱ ❈❛s❡ ✹✿ ❋❛r ✜❡❧❞✳

■♥✢✉❡♥❝❡ ♦❢ t❤❡ q✉❛❞r❛t✉r❡ ♦r❞❡r

❙❡t✉♣✿ ❙♣❤❡r✐❝❛❧ s❝❛tt❡r❡r✱ ✻✶✻ tr✐❛♥❣❧❡s❀ ❚✐♠❡ ✐♥t❡r✈❛❧✿ [✵, ✺]✱ ✷✵ t✐♠❡st❡♣s ❘❍❙✿ g(x, t) = ❝♦s(t − x✶) ❡−✻(t−x✶−✺)✷, x = (x✶, x✷, x✸)❚✳ ❘❡❢❡r❡♥❝❡ s♦❧✉t✐♦♥ φ❤✐❣❤ ✇❛s ❝♦♠♣✉t❡❞ ✇✐t❤ ns✐♥❣ = ✷✵✱ n♥❡❛r = ✶✺ ❛♥❞ n❢❛r = ✶✷✳ ns✐♥❣ n♥❡❛r n❢❛r r❡❧✳ L✷([✵, T], L✷(S))✲❡rr♦r ✶✵ ✽ ✻ ✶.✽✻ · ✶✵−✻ ✽ ✻ ✺ ✶.✷✻ · ✶✵−✺ ✻ ✺ ✹ ✽.✼✹ · ✶✵−✺ ✺ ✹ ✸ ✹.✺✽ · ✶✵−✹ ✺ ✸ ✸ ✶.✹✵ · ✶✵−✸ ✹ ✸ ✸ ✶.✽✶ · ✶✵−✸ ✹ ✸ ✷ ✷.✼✹ · ✶✵−✸

❚❛❜❧❡✿ ■♥✢✉❡♥❝❡ ♦❢ q✉❛❞r❛t✉r❡ ♦♥ t❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ ●❛❧❡r❦✐♥ ❛♣♣r♦①✐♠❛t✐♦♥

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts ♦♥ t❤❡ ❙♣❤❡r❡

10

−2

10

−1

10 10

−4

10

−3

10

−2

∆ t error p=0 p=1 ∆ t ∆ t1.5

✭❛✮ g(x, t) = t✻ ❡−✹t

10

−2

10

−1

10 10

−3

10

−2

10

−1

∆ t error p=0 p=1 p=2 ∆ t1.5 ∆ t2.5 ∆ t

✭❜✮ g(x, t) = t s✐♥(✸t)✷ ❡−t Y ✵

✶

❋✐❣✉r❡✿ ❈♦♥✈❡r❣❡♥❝❡ ♣❧♦ts ❢♦r T = ✶✱ ✷✺✻✽ tr✐❛♥❣❧❡s✱ ♣✐❡❝❡✇✐s❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ s♣❛❝❡✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts ♦♥ t❤❡ ❙♣❤❡r❡

5 10 15 20 25 30 35 40 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.6 Time t Galerkin solution Exact Solution Convolution quadrature solution

❋✐❣✉r❡✿ ●❛❧❡r❦✐♥ ❛♥❞ ❝♦♥✈♦❧✉t✐♦♥ q✉❛❞r❛t✉r❡ s♦❧✉t✐♦♥ ❝♦♠♣❛r❡❞ t♦ t❤❡ ❡①❛❝t s♦❧✉t✐♦♥ ❢♦r Γ = S✷ ❛♥❞ g(x, t) = t✹ ❡−✷t ✐♥ t❤❡ t✐♠❡ ✐♥t❡r✈❛❧ [✵, ✹✵]✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

−4 −3 −2 −1 1 2 −2 −1 1 2 −1 1 x y z P1=(0,0,0) P3=(0,0,1) P2=(−2,0,0) P4=(0,−2,0)

❋✐❣✉r❡✿ ❙❝❛tt❡r✐♥❣ ♦❢ ❛ ●❛✉ss✐❛♥ ♣✉❧s❡ ❢r♦♠ ❛ t♦r✉s ✇✐t❤ ♦❜s❡r✈❛t✐♦♥ ♣♦✐♥ts P✶, · · · , P✹✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

2 4 6 8 10 12 −2 −1 1

Time t 7 8 9 10 11 −0.05 0.05

✭❛✮ ❙♦❧✉t✐♦♥ u(P✶, t) ❛t P✶ = (✵, ✵, ✵).

2 4 6 8 10 12 −2 −1 1

Time t 7 8 9 10 11 −0.05 0.05

✭❜✮ ❙♦❧✉t✐♦♥ u(P✸, t) ❛t P✸ = (✵, ✵, ✶).

2 4 6 8 10 12 −1 −0.5 0.5

Time t

✭❝✮ ❙♦❧✉t✐♦♥ u(P✷, t) ❛t P✷ = (−✷, ✵, ✵).

2 4 6 8 10 12 −1 −0.5 0.5

Time t

✭❞✮ ❙♦❧✉t✐♦♥ u(P✹, t) ❛t P✹ = (✵, −✷, ✵).

❆ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t✐♦♥

❘❡❝❛❧❧ t❤❛t ✇❡ ✇❛♥t t♦ s♦❧✈❡ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠✿ ❋✐♥❞ φ s✳t✳ T

✵

• Γ

(S ˙ φ)(x, t)ζ(x, t)dΓxdt = T

✵

• Γ

˙ g(x, t)ζ(x, t)dΓxdt ❢♦r ❛❧❧ ζ✳ ❍❛✲❉✉♦♥❣ ✭✷✵✵✸✮ s❤♦✇❡❞ t❤❛t S ✐s ❛ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r ❢r♦♠ H− ✶

✷,− ✶ ✷ (Γ × [✵, T]) := L✷([✵, T], H− ✶ ✷ (Γ)) + H− ✶ ✷ ([✵, T], L✷(Γ)) ✐♥t♦

H

✶ ✷, ✶ ✷ (Γ × [✵, T]) := L✷([✵, T], H ✶ ✷ (Γ)) ∩ H ✶ ✷ ([✵, T], L✷(Γ)) ❛♥❞ ✐s t❤❡r❡❢♦r❡

❛♥ ♦♣❡r❛t♦r ♦❢ s♠♦♦t❤✐♥❣ t②♣❡ ✐♥ s♣❛❝❡ ❛♥❞ t✐♠❡✳ ■♥ ♦r❞❡r t♦ ♦❜t❛✐♥ ❛♥ ❛❞❛♣t✐✈❡ s❝❤❡♠❡ ✐♥ t✐♠❡ ✇❡ ❛♣♣❧② t❤❡ ❛ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t♦rs ♣r♦♣♦s❡❞ ❜② ❇✳ ❋❛❡r♠❛♥♥ ✭✷✵✵✵✮✳

❆ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t✐♦♥

❘❡❝❛❧❧ t❤❛t ✇❡ ✇❛♥t t♦ s♦❧✈❡ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠✿ ❋✐♥❞ φ s✳t✳ T

✵

• Γ

(S ˙ φ)(x, t)ζ(x, t)dΓxdt = T

✵

• Γ

˙ g(x, t)ζ(x, t)dΓxdt ❢♦r ❛❧❧ ζ✳ ❍❛✲❉✉♦♥❣ ✭✷✵✵✸✮ s❤♦✇❡❞ t❤❛t S ✐s ❛ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦r ❢r♦♠ H− ✶

✷,− ✶ ✷ (Γ × [✵, T]) := L✷([✵, T], H− ✶ ✷ (Γ)) + H− ✶ ✷ ([✵, T], L✷(Γ)) ✐♥t♦

H

✶ ✷, ✶ ✷ (Γ × [✵, T]) := L✷([✵, T], H ✶ ✷ (Γ)) ∩ H ✶ ✷ ([✵, T], L✷(Γ)) ❛♥❞ ✐s t❤❡r❡❢♦r❡

❛♥ ♦♣❡r❛t♦r ♦❢ s♠♦♦t❤✐♥❣ t②♣❡ ✐♥ s♣❛❝❡ ❛♥❞ t✐♠❡✳ ■♥ ♦r❞❡r t♦ ♦❜t❛✐♥ ❛♥ ❛❞❛♣t✐✈❡ s❝❤❡♠❡ ✐♥ t✐♠❡ ✇❡ ❛♣♣❧② t❤❡ ❛ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t♦rs ♣r♦♣♦s❡❞ ❜② ❇✳ ❋❛❡r♠❛♥♥ ✭✷✵✵✵✮✳

❆ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t✐♦♥

▲❡t ❛ s❡t ♦❢ ♠❡s❤♣♦✐♥ts Ξ := {tj}N−✶

j=✵ ✇✐t❤

✵ = t✵ < t✶ < . . . < tN−✷ < tN−✶ = T ❜❡ ❣✐✈❡♥✳ ▲❡t p = (p✶, p✷, · · · , pN) ∈ NN

✵ ❜❡ ❛ ❞❡❣r❡❡ ✈❡❝t♦r✳ ❲❡ ❞❡✜♥❡

V

p Ξ :=

• v : v =

N

• i=✶

pi

• m=✵

αi,mbi,m(t), αi,m ∈ R

• ❛♥❞

V ✵

Ξ := V (✵,...,✵) Ξ

. ❲❡ ❝♦♥s✐❞❡r ❛♥ ❛❜str❛❝t ❜♦✉♥❞❛r② ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠ Aξ = g ♦♥ [✵, T], ✇❤❡r❡ A ✐s ❛ ❜♦✉♥❞❡❞ ❛♥❞ ❜✐❥❡❝t✐✈❡ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦r ♦❢ ♦r❞❡r −✶✳

❚❤❡♦r❡♠

▲❡t A : H−✶/✷([✵, T]) → H✶/✷([✵, T]) ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠ ❛♥❞ ❧❡t G ⊂ H−✶/✷([✵, T]) ❜❡ ❛ tr✐❛❧ s♣❛❝❡ s❛t✐s❢②✐♥❣ V ✵

Ξ ⊂ G✳ ❋♦r ❛♥② s♦❧✉t✐♦♥

ξ ∈ H−✶/✷([✵, T]) ♦❢ t❤❡ ❜♦✉♥❞❛r② ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ❛♥❞ ❢♦r ❛❧❧ ❧♦❝❛❧ q✉❛s✐✲✉♥✐❢♦r♠ ♠❡s❤❡s Ξ ✇❡ ♦❜t❛✐♥ ❢♦r t❤❡ ●❛❧❡r❦✐♥ ❡rr♦r C❡✛

N

• ν=✶

η✷

ν ≤ ξ − ξ●❛❧❡r❦✐♥✷ H−✶/✷([✵,T]) ≤ Cr❡❧ N

• ν=✶

η✷

ν,

✇❤❡r❡ η✷

ν+✶ = |r|✷ H✶/✷([tν−✶tν+✶]) =

tν+✶

tν−✶

tν+✶

tν−✶

|r(t) − r(τ)|✷ |t − τ|✷ dτdt ✇✐t❤ r = Aξ●❛❧❡r❦✐♥ − g✳ C❡✛ ❛♥❞ Cr❡❧ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ξ, Ξ ❛♥❞ G✳

❆❞❛♣t✐✈❡ r❡✜♥❡♠❡♥t str❛t❡❣②✿

❈♦♠♣✉t❡ t❤❡ ❞✐s❝r❡t❡ s♦❧✉t✐♦♥ φ ∈ G✳ ❈♦♠♣✉t❡ t❤❡ ❡rr♦r ✐♥❞✐❝❛t♦rs {ηj}N

j=✶ ✭❛♣♣r♦①✐♠❛t❡❧②✮✳

■❢ N

j=✶ η✷ j ✐s s♠❛❧❧❡r t❤❛♥ ❛ ♣r❡s❝r✐❜❡❞ t♦❧❡r❛♥❝❡ → ❙t♦♣ t❤❡

r❡✜♥❡♠❡♥t✳ ❖t❤❡r✇✐s❡✱ ♠❛r❦ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ ∆ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❧❛r❣❡ ηj✳ ❉❡❝✐❞❡ ❢♦r ❡❛❝❤ ❡❧❡♠❡♥t ✐❢ ✐t s❤♦✉❧❞ ❜❡ h✲ ♦r p✲ r❡✜♥❡❞✳ ❘❡st❛rt t❤❡ ❝♦♠♣✉t❛t✐♦♥ ✇✐t❤ t❤❡ ❡♥r✐❝❤❡❞ ●❛❧❡r❦✐♥ s♣❛❝❡ ˜ G✳

❚❤❡ s♣❡❝✐❛❧ ❝❛s❡✿ Γ = S✷

▲❡t Γ ❜❡ t❤❡ ✉♥✐t s♣❤❡r❡✳ ❋✉rt❤❡r♠♦r❡ ❛ss✉♠❡ t❤❛t g(x, t) = g(t)Y m

n

✇❤❡r❡ Y m

n ❛r❡ t❤❡ s♣❤❡r✐❝❛❧ ❤❛r♠♦♥✐❝s✳ ❚❤✐s ❝❤♦✐❝❡ ❛❧❧♦✇s t♦ r❡❞✉❝❡ t❤❡

♦r✐❣✐♥❛❧ ♣r♦❜❧❡♠

• Γ

φ(y, t − x − y) ✹πx − y dΓy = g(x, t) ♦♥ Γ × [✵, T] t♦ t❤❡ ✉♥✐✈❛r✐❛t❡ ♣r♦❜❧❡♠ ✐♥ t✐♠❡ t

✵

L−✶(λn)(τ)φ(t − τ)dτ = g(t), t ∈ [✵, T], ✇❤❡r❡ λn ✐s ❛ ♣r♦❞✉❝t ♦❢ ♠♦❞✐✜❡❞ ❇❡ss❡❧ ❢✉♥❝t✐♦♥s✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 adaptive approximation equidistant approximation exact solution current adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = t✶.✺ ❡−t ❛♥❞ p = ✷✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 adaptive approximation equidistant approximation exact solution current adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = t✶.✺ ❡−t ❛♥❞ p = ✷✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 adaptive approximation equidistant approximation exact solution current adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = t✶.✺ ❡−t ❛♥❞ p = ✷✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 adaptive approximation equidistant approximation exact solution current adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = t✶.✺ ❡−t ❛♥❞ p = ✷✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 adaptive approximation equidistant approximation exact solution current adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = t✶.✺ ❡−t ❛♥❞ p = ✷✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 adaptive approximation equidistant approximation exact solution current adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = t✶.✺ ❡−t ❛♥❞ p = ✷✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 adaptive approximation equidistant approximation exact solution current adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = t✶.✺ ❡−t ❛♥❞ p = ✷✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

10 20 50 70 10

−5

10

−4

10

−3

10

−2

10

−1

10

degrees of freedom in time error

error of the adaptive approximation error of the equidistant approximation

❋✐❣✉r❡✿ ❈♦rr❡s♣♦♥❞✐♥❣ L✷✲❡rr♦rs✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 3.5 −60 −40 −20 20 40 60 80 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✳ ❢♦r g(t) = − s✐♥(✸✺t)t✸ ❡−✶✾✷(t−✶)✷ ❛♥❞ p = ✶✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 3.5 4 −60 −40 −20 20 40 60 80 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✳ ❢♦r g(t) = − s✐♥(✸✺t)t✸ ❡−✶✾✷(t−✶)✷ ❛♥❞ p = ✶✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 3.5 −60 −40 −20 20 40 60 80 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✳ ❢♦r g(t) = − s✐♥(✸✺t)t✸ ❡−✶✾✷(t−✶)✷ ❛♥❞ p = ✶✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 3.5 −60 −40 −20 20 40 60 80 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✳ ❢♦r g(t) = − s✐♥(✸✺t)t✸ ❡−✶✾✷(t−✶)✷ ❛♥❞ p = ✶✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 3.5 4 −60 −40 −20 20 40 60 80 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✳ ❢♦r g(t) = − s✐♥(✸✺t)t✸ ❡−✶✾✷(t−✶)✷ ❛♥❞ p = ✶✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 3.5 −60 −40 −20 20 40 60 80 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✳ ❢♦r g(t) = − s✐♥(✸✺t)t✸ ❡−✶✾✷(t−✶)✷ ❛♥❞ p = ✶✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 3.5 −60 −40 −20 20 40 60 80 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✳ ❢♦r g(t) = − s✐♥(✸✺t)t✸ ❡−✶✾✷(t−✶)✷ ❛♥❞ p = ✶✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 3.5 −60 −40 −20 20 40 60 80 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✳ ❢♦r g(t) = − s✐♥(✸✺t)t✸ ❡−✶✾✷(t−✶)✷ ❛♥❞ p = ✶✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 3.5 4 −60 −40 −20 20 40 60 80 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

Refinement

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✳ ❢♦r g(t) = − s✐♥(✸✺t)t✸ ❡−✶✾✷(t−✶)✷ ❛♥❞ p = ✶✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 3.5 4 −60 −40 −20 20 40 60 80 adaptive approx.

• equidist. approx.

exact solution

Refinement

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✳ ❢♦r g(t) = − s✐♥(✸✺t)t✸ ❡−✶✾✷(t−✶)✷ ❛♥❞ p = ✶✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 3.5 4 −60 −40 −20 20 40 60 80 adaptive approx.

• equidist. approx.

exact solution

Refinement

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✳ ❢♦r g(t) = − s✐♥(✸✺t)t✸ ❡−✶✾✷(t−✶)✷ ❛♥❞ p = ✶✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

10 10

1

10

2

10

−1

10 10

1

10

2

degrees of freedom in time error

error of the adaptive approximation error of the equidistant approximation

❋✐❣✉r❡✿ ❈♦rr❡s♣♦♥❞✐♥❣ L✷✲❡rr♦rs✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 −0.5 0.5 1 1.5 2 2.5 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = ♠❛①(✵, t − ✷) ❛♥❞ p = ✵✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 −0.5 0.5 1 1.5 2 2.5 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = ♠❛①(✵, t − ✷) ❛♥❞ p = ✵✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 −0.5 0.5 1 1.5 2 2.5 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = ♠❛①(✵, t − ✷) ❛♥❞ p = ✵✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 −0.5 0.5 1 1.5 2 2.5 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = ♠❛①(✵, t − ✷) ❛♥❞ p = ✵✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 −0.5 0.5 1 1.5 2 2.5 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = ♠❛①(✵, t − ✷) ❛♥❞ p = ✵✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 −0.5 0.5 1 1.5 2 2.5 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = ♠❛①(✵, t − ✷) ❛♥❞ p = ✵✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 −0.5 0.5 1 1.5 2 2.5 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = ♠❛①(✵, t − ✷) ❛♥❞ p = ✵✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 −0.5 0.5 1 1.5 2 2.5 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = ♠❛①(✵, t − ✷) ❛♥❞ p = ✵✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 −0.5 0.5 1 1.5 2 2.5 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = ♠❛①(✵, t − ✷) ❛♥❞ p = ✵✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

0.5 1 1.5 2 2.5 3 −0.5 0.5 1 1.5 2 2.5 adaptive approx.

• equidist. approx.

exact solution adaptive grid suggested grid

❋✐❣✉r❡✿ ❊①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥s ❢♦r g(t) = ♠❛①(✵, t − ✷) ❛♥❞ p = ✵✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts

10 10

1

10

2

10

−2

10

−1

10

degrees of freedom in time error

error of the adaptive approximation error of the equidistant approximation

❋✐❣✉r❡✿ ❈♦rr❡s♣♦♥❞✐♥❣ L✷✲❡rr♦rs✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts ✲ ❆ ✸❉ ❡①❛♠♣❧❡

❲❡ ❝♦♥s✐❞❡r ❛ s♣❤❡r✐❝❛❧ s❝❛tt❡r❡r ❞✐s❝r❡t✐③❡❞ ✐♥t♦ ✻✶✻ tr✐❛♥❣❧❡s✳ ❲❡ s❡t g(x, t) = −H(t − x✶ − ✷) (t − x✶ − ✷)✶.✺ (t − x✶ − ✷)✷ + ✺ ❢♦r x ∈ S✷ ❛♥❞ t ∈ [✵, ✷✺]✳ H(·) ❞❡♥♦t❡s t❤❡ ❍❡❛✈✐s✐❞❡ st❡♣ ❢✉♥❝t✐♦♥✳ ❲❡ st❛rt t❤❡ ❛♣❛t✐✈❡ ❛❧❣♦r✐t❤♠ ✇✐t❤ t❤❡ ❝♦❛rs❡ t✐♠❡ ❣r✐❞ ✺ ✵ ✺ ❛♥❞ ✉s❡❞ t❤❡ ♦❜s❡r✈❛t✐♦♥ ♣♦✐♥ts ✶ ✵ ✵ ❚ ✵ ✶ ✵ ❚ ✶ ✵ ✵ ❚ ❢♦r t❤❡ r❡✜♥❡♠❡♥t ✐♥❞✐❝❛t♦rs✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts ✲ ❆ ✸❉ ❡①❛♠♣❧❡

❲❡ ❝♦♥s✐❞❡r ❛ s♣❤❡r✐❝❛❧ s❝❛tt❡r❡r ❞✐s❝r❡t✐③❡❞ ✐♥t♦ ✻✶✻ tr✐❛♥❣❧❡s✳ ❲❡ s❡t g(x, t) = −H(t − x✶ − ✷) (t − x✶ − ✷)✶.✺ (t − x✶ − ✷)✷ + ✺ ❢♦r x ∈ S✷ ❛♥❞ t ∈ [✵, ✷✺]✳ H(·) ❞❡♥♦t❡s t❤❡ ❍❡❛✈✐s✐❞❡ st❡♣ ❢✉♥❝t✐♦♥✳ ❲❡ st❛rt t❤❡ ❛♣❛t✐✈❡ ❛❧❣♦r✐t❤♠ ✇✐t❤ t❤❡ ❝♦❛rs❡ t✐♠❡ ❣r✐❞ {✺ · l, l = ✵, . . . , ✺} ❛♥❞ ✉s❡❞ t❤❡ ♦❜s❡r✈❛t✐♦♥ ♣♦✐♥ts Ξ =

• (−✶, ✵, ✵)❚, (✵, ✶, ✵)❚, (✶, ✵, ✵)❚

❢♦r t❤❡ r❡✜♥❡♠❡♥t ✐♥❞✐❝❛t♦rs✳

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥ts ✲ ❆ ✸❉ ❡①❛♠♣❧❡

5 10 15 20 25 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 Time

Solution at (−1,0,0)T (adaptive) Solution at (−1,0,0)T (equidistant) Solution at (1,0,0)T (adaptive) Solution at (1,0,0)T (equidistant) Adaptive time grid Equidistant time grid

❋✐❣✉r❡✿ ❈♦♠♣❛r✐s♦♥ ❡q✉✐❞✐st❛♥t✴❛❞❛♣t✐✈❡ ❛♣♣r♦①✐♠❛t✐♦♥✳

❈♦♥❝❧✉s✐♦♥s ◆❡✇ t②♣❡s ♦❢ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s ❢♦r t❤❡ t✐♠❡ ❞✐s❝r❡t✐③❛t✐♦♥ ♦❢ t✐♠❡✲❞♦♠❛✐♥ ❜♦✉♥❞❛r② ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s

◮ ❙♠♦♦t❤ ❛♥❞ ❝♦♠♣❛❝t❧② s✉♣♣♦rt❡❞ ❜❛s✐s ❢✉♥❝t✐♦♥s ◮ ❙✐♠♣❧✐✜❡❞ ❣❡♥❡r❛t✐♦♥ ♦❢ t❤❡ s②st❡♠ ♠❛tr✐①✱ st❛♥❞❛r❞ s❝❤❡♠❡s ❝❛♥ ❜❡

✉s❡❞

❆❞❛♣t✐✈❡ r❡✜♥❡♠❡♥t ✐♥ t✐♠❡ ✉s✐♥❣ ❛ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t✐♦♥ ❈✉rr❡♥t ❛♥❞ ❢✉t✉r❡ ✇♦r❦ ❙♣❛❝❡✲t✐♠❡ ❛❞❛♣t✐✈❡ ❛❧❣♦r✐t❤♠ ❊✣❝✐❡♥t s♦❧✉t✐♦♥ ♦❢ t❤❡ r❡s✉❧t✐♥❣ ❧✐♥❡❛r s②st❡♠

❈♦♥❝❧✉s✐♦♥s ◆❡✇ t②♣❡s ♦❢ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s ❢♦r t❤❡ t✐♠❡ ❞✐s❝r❡t✐③❛t✐♦♥ ♦❢ t✐♠❡✲❞♦♠❛✐♥ ❜♦✉♥❞❛r② ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s

◮ ❙♠♦♦t❤ ❛♥❞ ❝♦♠♣❛❝t❧② s✉♣♣♦rt❡❞ ❜❛s✐s ❢✉♥❝t✐♦♥s ◮ ❙✐♠♣❧✐✜❡❞ ❣❡♥❡r❛t✐♦♥ ♦❢ t❤❡ s②st❡♠ ♠❛tr✐①✱ st❛♥❞❛r❞ s❝❤❡♠❡s ❝❛♥ ❜❡

✉s❡❞

❆❞❛♣t✐✈❡ r❡✜♥❡♠❡♥t ✐♥ t✐♠❡ ✉s✐♥❣ ❛ ♣♦st❡r✐♦r✐ ❡rr♦r ❡st✐♠❛t✐♦♥ ❈✉rr❡♥t ❛♥❞ ❢✉t✉r❡ ✇♦r❦ ❙♣❛❝❡✲t✐♠❡ ❛❞❛♣t✐✈❡ ❛❧❣♦r✐t❤♠ ❊✣❝✐❡♥t s♦❧✉t✐♦♥ ♦❢ t❤❡ r❡s✉❧t✐♥❣ ❧✐♥❡❛r s②st❡♠