Numerical analysis and computational solutionof integro-differential equations
Hermann Brunner
HONG KONG BAPTIST UNIVERSITY
and
MEMORIAL UNIVERSITY OF NEWFOUNDLAND
University of Strathclyde, 26 June 2013
1
Volterra integro-differential equations (VIDEs)
� Ordinary VIDEs (standard forms):
u′(t) + a(t)u(t) = f(t) +∫ t
0Kα(t, s)u(s)ds,
with u(0) = u0 and kernel
Kα(t, s) :=
⎧⎨⎩
(t − s)α−1K(t, s) if 0 < α ≤ 1
log(t − s)K(t, s) if α = 0.
↪→ “Neutral” VIDEs:
u(k)(t) +k−1∑j=0
aj(t)u(j)(t) = f(t) +
k∑j=0
∫ t
0
Kαj(t, s)u(j)(s)(t)ds
(k ≥ 1, 0 ≤ αj ≤ 1), with u(j)(0) = uj (j = 0,1, . . . , k − 1).
↪→ Nonlinear VIDEs (Hammerstein form):
u′(t) = f(t,u(t)) +
∫ t
0
Kα(t, s)G(s,u(s))ds.
2
� Non-standard ordinary VIDEs:
In the mathematical modelling of memory effects, VIDEs often occur innon-standard form:
↪→ Predator-prey model (Volterra, 1927):
dN1(t)
dt= N1(t)
(ε1 − γ1N2(t) −
∫ t
t−τ
F1(t − s)N1(s)ds
)dN2(t)
dt= N2(t)
(−ε2 + γ2N1(t) −
∫ t
t−τ
F2(t − s)N2(s)ds
).
↪→ Stretching of polymers (Lodge et al., 1978; Markowich & Renardy, 1983):
εu′(t) = b(t)uβ(t) +
∫ t
−∞k(t − s)G(u(t),u(s))ds,
where 0 < ε � 1 , and β = 2 (filament) or β = 1/2 (sheet).
↪→ Auto-convolution VIDEs (theory of turbulence):
u′(t) + a(t)u(t) = f(t) +
∫ t
0
K(t, s)u(t − s)u(s)ds
(v. Wolfersdorf & Janno, 2009/2011).3
(Non-standard VIDEs / contd.)
↪→ Neutral VIDEs:
d
dt
(A0u(t) −
∫ 0
−τA1(s)u(t + s)ds
)= B0u(t) + f(t)
(t > 0, τ > 0), with Ai, Bi ∈ Rd (d = 8), detA0 = 0 (typically, the last row of A0
is a row of zeros):
Elastic motions of 3-degree-of-freedom airfoil with flap in 2D incompressible flow
(Burns et al., 1983+).
↪→ Integro-differential algebraic equations:
A(t)u′(t) + B(t)u(t) = f(t) +∫ t
0Kα(t, s)u(s)ds,
with u ∈ Rd (d ≥ 2), A(·), B(·) ∈ Rd×d, Kα(·, ·) ∈ Rd×d, and
detA(t) = 0 on I, with rankA(t) > 0 :
Dolezal, 1960s (Kirchhoff laws for electrical networks); Bulatov & Chistyakov, 1997;
Bulatov & Chistyakova, 2011.4
� Partial VIDEs (standard forms)
Parabolic VIDEs:
ut + Au = f +∫ t
0B(t, s)u(s, ·)ds, t ∈ I, x ∈ Ω ⊂ Rd,
with u(0, x) = 0, x ∈ Ω; u(t, x) = 0, (t, x) ∈ I × ∂Ω, and where
A is a linear, uniformly elliptic (spatial) differential operator,
B is a linear spatial differential operator of order ≤ 2.
Hyperbolic VIDEs:
utt + Au = f +∫ t
0B(t, s)u(s, ·)ds, t ∈ I, x ∈ Ω ⊂ Rd,
with u(0, x) = u0(x), ut(0, x) = u1(x) (x ∈ Ω); u(t, x) = 0 on I × ∂Ω.
Partial VIDEs with positive memory:
ut +∫ t
0kα(t − s)Au(s, ·)ds = f (0 < α < 1),
with∫ T
0
∫ t
0kα(t − s)ϕ(s)ϕ(t) ds dt ≥ 0 for all ϕ ∈ C(I), and kα(z) = zα−1/Γ(α).
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Remark: The partial VIDE
ut = f +∫ t
0k(t − s)Δu(s, ·)ds
interpolates between the linear diffusion equation and the linearwave equation:
k(t − s) = δ(t − s) ⇒ ut = f + Δu.k(t − s) = 1 ⇒ utt = ft + Δu.
The partial Volterra integral equation
u = f +∫ t
0
(t − s)α−1
Γ(α)Δu(s, ·)ds (1 < α < 2)
possesses the analogous property (Fujita (1990)).
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� Non-standard partial VIDEs
↪→ Elliptic VIDEs:
Δu +∫ t
0
3∑j=1
aj(t, s)∂2u(s, ·)
∂x2j
ds = f ,
where u = u(t,x), t ≥ 0, x ∈ Ω ⊂ R3 :
Models viscoelastic materials with memory (Volterra, 1908/1912; see also
Shaw & Whiteman, 1996).
↪→ Fractional evolution equations:
ut = f + BαAu (e.g. A = Δ ),where either
(Bαv)(t) =∂
∂t
∫ t
0
(t − s)α
Γ(1 + α)v(s)ds, −1 < α < 0,
or
(Bαv)(t) =
∫ t
0
(t − s)α−1
Γ(α)v(s)ds, 0 < α < 1 :
α ∈ (−1,0) : slow or anomalous diffusion in fractured media;
α ∈ (0,1) : wave propagation in viscolealstic materials.
7
� Vito Volterra in his 1909 paper on IDEs:
“The problem of solving integro-differential equations constitutesin general a problem that differs essentially from the problems of solving
differential equations and the usual ones for integral equations”.
(“Il problema della risoluzione delle equazioni integro-differenzialicostituisce in generale un problema essenzialmente distinto dai problemidelle equazioni differenziali e da quelli ordinarii delle equazioni integrali.” )
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Outline
� Time-stepping for ordinary VIDEs(Runge-Kutta and collocation methods; DG methods; hp versions)
� Time-stepping for partial VIDEs(Runge-Kutta and collocation methods; hp-DG and discretised DG methods)
� Computational aspects(Fractional diffusion and wave equations; unbounded spatial domains; IDEs
of Fredholm type)
� Computational challenges in solving IDEs
� Ongoing and future work
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Collocation and Galerkin spaces (time-stepping)
Let Ih := {tn : 0 = t0 < t1 · · · < tN = T} be a mesh for the given interval
I := [0,T] , with en := (tn, tn+1), hn := tn+1 − tn, h := max(n){hn}.
• Globally continuous piecewise polynomials (m ≥ 1):
S(0)m (Ih) := {v ∈ C(I) : v|en
∈ Pm (0 ≤ n ≤ N − 1)},where Pm = Pm(en) is the space of polynomials of degree m .
• Discontinuous piecewise polynomials (m ≥ 0):
S(−1)m (Ih) := {v : v|en
∈ Pm (0 ≤ n ≤ N − 1)}.
• hp-spaces: For given degree vector m := (m0,m1, . . . ,mN−1 ), let
S(0)m (Ih) := {v ∈ C(I) : v|en ∈ Pmn
(0 ≤ n ≤ N − 1)}and
S(−1)m (Ih) := {v : v|en
∈ Pmn(0 ≤ n ≤ N − 1)}.
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Galerkin and collocation methods for
u′(t) + a(t)u(t) = f(t) + (Vαu)(t), t ∈ I = [0, T ],
where (Vαu)(t) =
∫ t
0
kα(t − s)K(t, s)u(s)ds, with K ∈ C(D) and
kα(t − s) :=
{(t − s)α−1 if 0 < α ≤ 1,log(t − s) if α = 0.
• Exact continuous Galerkin method: uh ∈ S(0)m (Ih) so that
〈u′h + auh, φ〉 = 〈f , φ〉 + 〈Vαuh, φ〉 for all φ ∈ S(0)
m (Ih) ,
with uh(0) = u(0) .• Exact discontinuous Galerkin method: uh ∈ S(−1)
m (Ih) so that
〈u′h + auh, φ〉 = 〈f , φ〉 + 〈Vαuh, φ〉 for all φ ∈ S(−1)
m (Ih) .
(↪→ Note: The inner products 〈·, ·〉 (integrals) are assumed to be known exactly.)
• Continuous collocation method: uh ∈ S(0)m (Ih) so that
u′h(t) + a(t)uh(t) = f(t) + (Vαuh)(t), t ∈ Xh ,
with uh(0) = g(0) and Xh := {tn + cihn : 0 < c1 < · · · < cm ≤ 1 (0 ≤ n ≤ N − 1)}.
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DG equation based on the variational form of
u′(t) + a(t)u(t) = f(t) + (Vαu)(t),
with a(t) ≥ a0> 0 and
(Vαu)(t) =
∫ t
0
(t − s)α−1K(t, s)︸ ︷︷ ︸=:Kα(t,s)
u(s)ds (0 < α ≤ 1) :
BDG(uh,v) = FDG(v) for all v ∈ S(−1)m (Ih),
where
BDG(uh,v) :=N−1∑j=0
∫ej
[u′h(t) + a(t)uh(t)]v(t)dt
−N−1∑j=0
∫ej
(∫ t
0
Kα(t, s)uh(s)ds
)v(t)dt +
N−1∑j=0
[uh]jv+j + uh(0
+)v(0+),
FDG(v) := u0v(0+) +N−1∑j=0
∫ej
f(t)v(t)dt,
with [uh]j := uh(t+j ) − uh(t
−j ) .
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Ordinary VIDEs:
u′(t) + a(t)u(t) = f(t) + (Vαu)(t),
with a(t) ≥ a0> 0 and (Vαu)(t) =
∫ t
0
Kα(t, s)u(s)ds (0 < α ≤ 1) :
Time-stepping form of the (exact) DG equation:
For known uh on ej (j ≤ n − 1), find uh|en∈ Pm, with initial value U−
n−1, so that∫en
[u′h(t) + a(t)uh(t)]φ(t)dt + U+
n φ+n −
∫en
(∫ t
tn
Kα(t, s)uh(s)ds
)φ(t)dt
= U−n φ+
n +
∫en
f(t)φ(t)dt+
∫en
(∫ tn
0
Kα(t, s)uh(s)ds
)︸ ︷︷ ︸
history of uh on (0,tn)
φ(t)dt
for all φ ∈ Pm and n = 0,1, . . . , N − 1. Here, U+n := uh(t+n ) and U−
n := uh(t−n ) .
13
Attainable orders of convergence
� Collocation in S(0)m (Ih) (0 < c1 < · · · < cm ≤ 1)
↪→ Smooth solutions (u ∈ Cd(I), d ≥ m + 1):
‖u − uh‖∞ ≤ Chm+1 (if∫ 1
0
m∏i=1
(s − ci)ds = 0).
↪→ Non-smooth solutions (u ∈ Cm+1(0,T], u′′(t) ∼ γt−α as t → 0+):
On uniform meshes: ‖u − uh‖∞ ≤ Ch2−α (for all m ≥ 1) .
On graded meshes (tn = (n/N)rT, n = 0, . . . , N ; r > 1) :
‖u − uh‖∞ ≤ CN−(m+1−α) if r ≥ (m + 1 − α)/(2 − α)
(Tang, 1992; B., Pedas & Vainikko, 2001).� hp-DG in S(−1)
m (Ih) (mi = m for all ):
‖u − uh‖∞ ≤ C
{m−shmin{s,m}+1 ‖u‖Ws+1,∞ if u ∈ Ws+1,∞(I),e−βm if u is analytic on I,
with C > 0, β > 0 independent of m (Brunner & Schotzau, 2006).
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� hp-DG / non-smooth solutions:
u′(t) + a(t)u(t) = f(t) + (Vαu)(t) (0 < α < 1)
with f(t) = f0(t) + tβf1(t) (β > 0, β �∈ N).
↪→ Geometric local mesh refinement (Extension of Schotzau & Schwab, 2000):(i) Choose a (uniform) hp-discretisation Ih of I = [0,T];(ii) choose a geometric submesh on e0 = (0, t1): for given grading factorσ = σ(α, β) ∈ (0,1), let
t0,0 := 0, t0,μ := σr+1−μt1 (1 ≤ μ ≤ r + 1).
The locally refined hp-discretisation of I is denoted by Ih(r, σ), and thecorresponding hp-DG space is S(−1)
m (Ih(r, σ)).
↪→ (B. & Schotzau, 2006) There exists a (linear) degree vector for Ih(r, σ),
m := (m0,1, . . . ,m0,r+1︸ ︷︷ ︸on e0
; m1, . . . , mN−1︸ ︷︷ ︸on e1,...,eN−1
),
and a grading factor σ so that the hp-DG solution uh ∈ S(−1)m (Ih(r, σ)) satisfies
‖u − uh‖∞ ≤ Ce−bM1/2,
with positive constants C, b and M := dim S(−1)m (Ih(r, σ)).
15
� h- and hp-DG time-stepping for parabolic VIDEs:
ut − Δu = f +∫ t
0Kα(t, s)Bu(s, ·)ds (0 < α < 1).
↪→ Larsson, Thomee & Wahlbin (1998): h-version for S(−1)m (Ih) with m = 0,1.
↪→ Mustapha, B., Mustapha & Schotzau (2011): hp-version with arbitrary m ≥ 0:continuous Galerkin in space; DG time-stepping using initial local geometricmesh refinement and linearly increasing polynomial degree vectors.
⇒ Error estimates that are explicit in the time-steps hn, the polynomial degrees mi,and the regularity parameters of the exact solution u.
If u has start-up singularities but is analytic for t > 0 ⇒ exponential convergence
rates.
Open problem: hp-DG convergence analysis if u has singularitiescaused by non-smooth initial data ?
16
Exact versus discretised DG methods
• Exact DG equation: uh ∈ S(−1)m (Ih) so that
〈u′h + auh, φ〉 = 〈f , φ〉 + 〈Vαuh, φ〉 for all φ ∈ S(−1)
m (Ih) .
↪→ Approximation of inner products, e.g.
〈auh, φ〉en=
∫en
a(s)uh(s)φ(s)ds = hn
∫ 1
0
a(tn + shn)uh(tn + shn)φ(tn + shn)ds,
by (interpolatory) numerical quadrature (with 0 ≤ d0 < · · · < dq ≤ 1 ):
〈auh, φ〉en≈ hn
q∑j=0
wja(tn + djhn)uh(tn + djhn)φ(tn + djhn) .
• Discretised DG for ODE (Vα = 0). If q = m :Collocation-based (m + 1)-stage implicit Runge-Kutta method with jump discontinuity
terms (Lasaint & Raviart, 1974; Delfour et al., 1981; see also Estep & Stuart (2001) for
dissipativity-preserving discretisations).• Discretised CG for ODE (using abscissas 0 ≤ d1 < · · · < dm ≤ 1 ):
Collocation-based m-stage continuous implicit Runge-Kutta method, with collocation
points ci = di (i = 1, . . . , m) given by the quadrature abscissas (Hulme, 1972).
17
Discretised DG equation for VIDE: Assume uh ∈ S(−1)m−1(Ih)
• Time-stepping form of the exact DG equation: Find uh on en so that∫en
(u′
h(t) + a(t)uh(t))φ(t)dt + U+
n φ+n −
∫en
(∫ t
tn
Kα(t, s)uh(s)ds
)φ(t)dt
= U−n φ+
n +
∫en
f(t)φ(t)dt+
∫en
(∫ tn
0
Kα(t, s)uh(s)ds
)︸ ︷︷ ︸
history of uh on (0,tn)
φ(t)dt
for all φ ∈ Pm−1 and n = 0,1, . . . , N − 1.
↪→ Approximation of inner products over en: if 0 ≤ d1 < · · · < dm ≤ 1 are given,∫en
F(t)φn,i(t)dt ≈ hn
m∑j=1
bjF(tn + djhn)φn,i(tn + djhn) = hnbiF(tn + dihn)
for all local Lagrange basis functions {φn,i} with respect to the {dj}.
18
⇒ Discretised DG equation for uh ∈ S(−1)m−1(Ih):
u′h(t) + a(t)uh(t) + b−1
i U+n φ+
n,i−∫ t
tn
Kα(t, s)uh(s)ds
= b−1i U−
n φ+n,i + f(t)+
n−1∑ =0
∫ t +1
t
Kα(t, s)uh(s)ds︸ ︷︷ ︸history of uh on (0,tn)
,
i = 1, . . . ,m, with t = tn + dihn (n = 0, . . . , N − 1).
• Comparison: Collocation solution wh ∈ S(0)m (Ih) on en :
w′h(t) + a(t)wh(t) − hn
∫ ci
0
Kα(t, tn + shn)wh(tn + shn)ds
= f(t) +n−1∑ =0
∫ t +1
t
Kα(t, s)wh(s)ds︸ ︷︷ ︸history of wh on [0,tn]
for t = tn + cihn (i = 1, . . . , m; n = 0, . . . , N − 1), with uh(0) = u0 .
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� Discretised CG time-stepping for VIDEs:
Approximation of inner products (on en) by interpolatory m-point quadrature
formula using the abscissas {tn + djhn} with 0 ≤ d1 < · · · < dm ≤ 1 :
↪→ m-point collocation solution in S(0)m (Ih) with ci = di (≡ m-stage continuous
implicit Volterra-Runge-Kutta method, with ci = di as Runge-Kutta abscissas).
↪→ Ordinary VIDEs: Brunner (2004: Ch. 3/Ch. 6)
↪→ Parabolic VIDEs (implicit Volterra-Runge-Kutta-methods): Brunner, Kauthen
& Ostermann (1995).
20
Computational aspects� Fractional diffusion and wave equations:
ut = f + BαAu (e.g. A = Δ),
where either
(Bαv)(t) =∂
∂t
∫ t
0
(t − s)α
Γ(1 + α)v(s)ds, −1 < α < 0,
or
(Bαv)(t) =
∫ t
0
(t − s)α−1
Γ(α)v(s)ds, 0 < α < 1 .
↪→ Convolution quadrature and Laplace transform techniques:
Lubich, Sloan & Thomee (1996), Cuesta, Lubich & Palencia (2006), Lopez-Fernandez,
Lubich & Schadle (2008), Xu (2003,2008), McLean & Thomee (2010).
↪→ DG time-stepping:
Mustapha & McLean (2009,2011), McLean (2012); Mustapha & McLean (2013)
(superconvergence); Mustapha & Ryan (2013) (postprocessing of DG solutions).↪→ DG with adaptive time-stepping:
Adolfsson, Enelund & Larsson (2003), Brunner, Ling & Yamamoto (2010).
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(Computational aspects / contd.)
Elliptic partial VIDEs
� The elliptic partial VIDE for u = u(t,x),
Δu +
∫ t
0
3∑j=1
aj(t, s)∂2u(s, ·)
∂x2j
ds = f
(t ≥ 0, x ∈ Ω ⊂ R3 ), was introduced and studied by Volterra in 1908/1912.
General form:
Au +∫ t
0B(t, s)u(s, ·)ds = f (t ∈ [0, T ], x ∈ Ω ⊂ Rd),
with A self-adjoint and strongly elliptic.
↪→ Computational solution:
Use of CG in space ( S(0)1 (Ωh)) and DG in time ( S(−1)
0 (Iτ)); a posteriori error
estimation: Shaw & Whiteman (1996).
22
(Computational aspects / contd.)
� Unbounded spatial domains Ω ⊂ Rd :
ut − Δu = f +∫ t
0K(t − s, ·)u(s, ·)ds, x ∈ Ω, t ≥ 0,
where Ω ⊂ R2 is an infinite strip: Construction of artificial boundary conditions
for bounded computational domain (Han, Zhu, B. & Ma, 2006).
ut − Δu =∫ t
0k(t − s)up(s, ·)ds, x ∈ Ω, t ≥ 0, (1)
with Ω = R2 or Ω = cone in R2 , and p > 1: solution blows up in finite time.↪→ For k(t − s) = δ(t − s) ( ⇒ PDE: ut − Δu = up ):Artificial (nonlinear) boundary conditions; adaptive time-stepping based on
one-point collocation (Zhang, Han & Brunner, 2011).↪→ For general k(t − s) > 0 ( ⇒ partial VIDE (1)):Computational solution (e.g. collocation or DG time-stepping on the bounded
computational domain with artifical boundary conditions): Open problem.
(� PDEs: ↪→ 2013 book, Artificial Boundary Method by Han & Wu.)
23
(Computational aspects / contd.)
Parabolic IDEs of Fredholm-type
� The linear Fredholm IDE
ut + Au = 0 on I × Ω
(I = [0, T ], Ω ⊂ Rd bounded), where the non-local operator A is the sum of a
second-order (elliptic) differential operator and a Fredholm-type integral operator,
arises in option pricing under Levy processes (Jacob, 2005).
↪→ Computational solution: Wavelet discretisation in space (+ wavelet compression
techniques for densely populated matrices) and DG time-stepping: Matache, Schwab
& Wihler (2006).
But: For many classes of (semilinear) Fredholm IDEs the design and the
analysis of computational methods remain to be studied (see next slide).
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� Computational challenges:Numerical detection of blow-up / computation of blow-up time for semilinear
Fredholm IDEs arising in
• Chemical reaction processes:
ut − Δu =∫Ω
F(u(·,y))dy,
(with t ∈ [0, T ], x ∈ Ω � Rd (where Ω = Rd or Ω � Rd is bounded)), describes
chemical reaction processes where solutions may blow up in finite time.
↪→ Chadam & Yin (1989/1993), Chadam, Peirce & Yin (1992); Souplet (1998).
• Reactive-diffusive ignition models:
ut − Δu = F(u) + [vol (Ω)]−1∫Ω
ut(·,y)dy.
↪→ Bebernes et al. (1982+).• Non-local reaction-diffusion equations:
ut − Δu =∫ t
0
∫Ωk(t − s)H(·,y)G(u(s,y))dy ds,
with (e.g.) G(u) = up, p > 1. ↪→ Souplet (1998).
25
Ongoing / future work:
• Extension of a posteriori error estimation for hp-DG solutions forparabolic PDEs (Schotzau & Wihler, 2010) to discretised hp-DGtime-stepping for ordinary VIDEs and parabolic partial VIDEs.
• hp-collocation (CC) time-stepping for ordinary VIDEs and parabolic VIDEs,including problems with non-smooth solutions.
• A priori and a posteriori error estimation for hp-collocation (CC) time-stepping for ordinary and parabolic VIDEs.
• hp-collocation time-stepping for fractional diffusion and wave equations ?
• Analysis of time-stepping methods (DG / collocation) for parabolic VIDEs onunbounded spatial domains ?
• Semilinear partial Volterra and Fredholm IDEs with blow-up solutions:numerical detection of blow-up (using, e.g., moving mesh methods); a posteriorierror estimates for blow-up time ?
• Computational methods for VIDEs with highly oscillatory kernels ?
(↪→ Slides / References: www.math.hkbu.edu.hk/∼hbrunner/)
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↪→ With thanks to Arieh for the ‘colours’ . . . !
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