Modeling the Di usion of Heat Energy within Composites of ...Modeling the Di usion of Heat Energy within Composites of Homogeneous Materials using the Uncertainty Principle Elyse M

Computational and Applied Mathematics manuscript No.(will be inserted by the editor)

Modeling the Diffusion of Heat Energy withinComposites of Homogeneous Materials using theUncertainty Principle

Elyse M. Garon · James V. Lambers

Received: date / Accepted: date

Abstract The goal of this paper is to develop a highly accurate and effi-cient numerical method for the solution of a time-dependent partial differen-tial equation with a piecewise constant coefficient, on a finite interval withperiodic boundary conditions. The resulting algorithm can be used, for exam-ple, to model the diffusion of heat energy in one space dimension, in the casewhere the spatial domain represents a medium consisting of two homogeneousmaterials. The resulting model has, to our knowledge, not yet been solved inclosed form through analytical methods, and is difficult to solve using existingnumerical methods, thus suggesting an alternative approach.

The approach presented in this paper is to represent the solution as alinear combination of wave functions that change frequencies at the interfacesbetween different materials. It is demonstrated through numerical experimentsthat by using the Uncertainty Principle to construct a basis of such functions,in conjunction with a spectral method, a mathematical model for heat diffusionthrough different materials can be solved much more efficiently than withconventional time-stepping methods.

Keywords heat equation · Uncertainty Principle · interface problem ·spectral methods

1 Introduction

The purpose of this paper is to numerically solve the heat equation in onespace dimension, such as within a rod, in the case where the heat flow is

E. M. Garon, J. V. LambersDepartment of Mathematics, The University of Southern Mississippi, 118 College Dr #5045,Hattiesburg, MS 39406, USATel.: +1-601-2664289Fax: +1-601-2665818E-mail: [email protected]

2 Elyse M. Garon, James V. Lambers

through a medium consisting of two homogeneous materials. For example, ifthe medium is a rod, the rod could be made of a copper rod and a silver rodwelded together, as depicted in Figure 1. The figure illustrates the heat flowthrough an insulated rod made of two homogeneous materials, and the endsof the rod are immersed in mediums held at a fixed temperature at each end.

Fig. 1 Heat flow through composites of homogeneous materials

The heat equation used to model the diffusion of heat through a singlehomogeneous material in one space dimension is as follows (Farlow, 1993):

ut = α2uxx (1)

where

ut = rate of change in temperature with respect to time

α2 = diffusivity

uxx = concavity of the temperature curve

This equation, also known as the diffusion equation, is useful for modeling theevolution in time of the quantity u that represents a density of, for example,heat energy or chemical concentration (Evans, 1994).

Equation (1) is a second order parabolic partial differential equation (PDE)that can be solved using a variety of analytical and numerical techniques (Far-low, 1993; Gustafsson et al., 1995). However, the challenge of creating a math-ematical model for heat diffusion through a medium consisting of two or morehomogeneous materials is that the heat equation will be modeled using a piece-wise constant function α2(x). The equation then becomes:

ut = (α2(x)ux)x. (2)

In this paper, we investigate the numerical solution of (2) on a finite inter-val, with periodic boundary conditions imposed in order to focus on the be-havior at the interface between materials, rather than on boundary effects. Wealso include evidence from numerical experiments that the approach describedin this paper can be applied to problems with different boundary conditions. Itis hypothesized that through separation of variables, the solution can be repre-sented as a linear combination of eigenfunctions that are wave functions whichchange frequencies at the boundaries of different materials. The UncertaintyPrinciple provides guidance on how to find these eigenfunctions. With these

Modeling Diffusion using the Uncertainty Principle 3

approximate eigenfunctions, the solution can be represented in such a waythat numerical solution of an equation with a piecewise constant coefficientis almost as simple as that of an equation with a constant coefficient, usingan eigenfunction expansion. Although this paper applies this approach to theheat equation, it can also be used with other PDE, such as the second-orderwave equation.

The outline of the paper is as follows: Section 2 discusses the difficultieswith existing methods for this type of diffusion problem. Section 3 describesthe methodology that will be used to solve the problem. Section 4 will presentthe details of a practical algorithm for computing eigenfunctions of the spatialdifferential operator. Section 5 will describe how these eigenfunctions can beused to solve the PDE. Numerical results are presented in Section 6, andconclusions and directions for future work are given in Section 7.

2 Literature Review

When solving a linear, homogeneous PDE with a constant coefficient on afinite interval analytically, the solution can be represented as a series of sinesand/or cosines because they are eigenfunctions of the spatial differential oper-ator α2(∂/∂x)2. However, when the coefficient is not constant, this approachis not viable because the eigenfunctions are unknown, except in special casessuch as the Lagrange, Bessel, or Hermite equations (Arfken et al., 2012).

The problem of solving PDEs with discontinuous coefficients is well-posedand theoretically well-known (Aronson, 1968; Ladyzenskaja et al., 1968). Thesolution of (2) can be found analytically in free space (Lejay and Pichot, 2012;Walsh, 1978); furthermore, for Dirichlet or Neumann boundary conditions, ananalytical solution can be found using the classical layer potential (Folland,1995; Portenko, 1990), though such a solution is not in closed form. Thereare a variety of Monte Carlo methods that address this problem (Etore, 2006;Hoteit et al., 2002; LaBolle et al., 1996; Lejay and Pichot, 2012; Uffink, 1985),and in addition, there are deterministic finite element and finite differencemethods that have also been employed (Gerardo-Giorda et al., 2004; Zunino,2003).

With numerical methods, difficulties arise because the more continuousderivatives a function has, the more rapidly its Fourier series converges (Gustafs-son et al., 1995), but in this case, rapid convergence does not occur due to thediscontinuity in the coefficient, which leads to a discontinuous first derivativein the solution. As a result, solutions will have non-negligible high frequencycomponents (Gustafsson et al., 1995). It follows that higher spatial resolutionis required to represent the solution to high accuracy, which poses difficultiesfor both implicit and explicit time-stepping methods due to stiffness. Stiffnessoccurs when solutions have both low- and high-frequency components that arecoupled together and cannot be computed independently of one another, be-cause they cannot be separated (Burden and Faires, 2010). Because of stiffness,the highest-frequency component in the solution forces the time step used in


numerical methods to be very small, even though the high-frequency compo-nents make a negligible contribution to the solution. This time step constraintis due to the CFL condition, which indicates how small the time step mustbe relative to the space step (Gustafsson et al., 1995). Because more, smallertime steps must be taken, the computational effort increases substantially. Inview of these issues that arise in existing numerical methods, it is worthwhileto consider an alternative approach.

The alternative approach considered in this paper involves computing highlyaccurate approximate eigenvalues and eigenfunctions of the spatial differentialoperator. A similar eigenvalue problem involving a discontinuous coefficient,but with Dirichlet boundary conditions, was described and solved in (Filocheet al., 2012); however, that work does not include a practical numerical methodfor solving the equations that characterize the eigenfunctions. For the prob-lem considered in this paper, with periodic boundary conditions, the SAKprinciple, derived from the Uncertainty Principle by Fefferman (Fefferman,1983), will be used to develop such a practical algorithm for computing theeigenvalues and eigenfunctions, as described in detail in the next two sections.

3 Methodology

To solve the heat equation (2) on a finite interval with periodic boundary condi-tions, a numerical method will be designed using approximate eigenfunctions.In contrast with constant-coefficient problems, in which the eigenfunctions aresines or cosines with fixed frequencies, the eigenfunctions will change frequencyat the interface between materials.

The Uncertainty Principle suggests, at least indirectly, how to find thesefrequency changes. In its original form due to Heisenberg, the UncertaintyPrinciple states that the position and momentum of a particle cannot simulta-neously be measured with arbitrarily high precision (Heisenberg, 1927). Mathe-matically, this is equivalent to the impossibility of a function f and its Fouriertransform f both being concentrated (i.e. supported) within an arbitrarilysmall box in phase space, where phase space is the Cartesian product of thephysical space and the frequency space (Fefferman, 1983). Similar results werealso stated in (Amrein and Berthier, 1977; Benedicks, 1985).

The Uncertainty Principle has been used for eigenvalue-counting with self-adjoint differential operators (Fefferman, 1983). A straightforward interpreta-tion of the Uncertainty Principle in this context is that the number of eigen-values of such an operator

A(x,D) =∑|j|≤m

aj(x)Dj , D =1

i

∂

∂x, (3)

where

i =√−1, Dj =

1

ij∂j

∂xj,


that are less than some threshold K is approximately equal to the volume ofthe subset of phase space

S(A,K) = {(x, ω)|A(x, ω) < K}, (4)

whereA(x, ω) = e−iωxA(x,D)eiωx =

∑|j|≤m

aj(x)ωj (5)

is the symbol of the operator A(x,D).The rationale behind this interpretation is that any function, including an

eigenfunction of A(x,D), can be concentrated only within a subset of phasespace of volume at least 1, in view of the Uncertainty Principle, so the volumeof S(A,K) provides an upper bound on how many eigenfunctions, which areorthogonal due to A(x,D) being self-adjoint, can be concentrated into disjointsubsets of S(A,K). While this upper bound has been shown to be asymptot-ically correct for elliptic operators as K → ∞ (Carleman, 1960; Weyl, 1950),it can be grossly inaccurate even for simple operators (Fefferman, 1983). IfS(A,K) consists of several disconnected subsets of phase space that are ofsmall volume, then no eigenfunction can be concentrated within it. As such, itis possible that simple volume-counting may estimate that many eigenvaluesare less than K, when in fact there may be none (Fefferman, 1983).

Fefferman proposed a more faithful interpretation of the Uncertainty Prin-ciple, called the SAK principle, which states that the number of eigenvaluesof A(x,D) that are less than K is approximately equal to the number of “dis-torted unit cubes” (that is, subsets that can be mapped to unit cubes bycanonical transformations, which preserve volume) that can be packed in theset S(A,K). Based on the SAK principle, phase space can be divided intoregions into which an eigenfunction can “fit”, or can be mostly concentratedwithin. Furthermore, if a function φ(x) is mostly concentrated in phase spacein such a distorted unit cube, centered at (x0, ω0), then applying the operatorA(x,D) to this function yields

A(x,D)φ(x) ≈ A(x0, ω0)φ(x) (6)

meaning that A(x0, ω0) is an approximate eigenvalue, and therefore φ(x) is anapproximate eigenfunction (Fefferman, 1983).

For the crude approximation (6) to be more accurate, the symbol A(x, ω)of the operator A(x,D) should be nearly constant within the distorted unitcube. It therefore makes sense that we can find approximate eigenfunctionsof the operator by studying the level curves of the symbol. The level curvesjump at points of discontinuities, which suggest the changes in frequency.This phenomenon can be seen in Figure 2 and Figure 3 where the continuousfrequency of the constant coefficient can be seen in Figure 2, and jumps infrequencies can be clearly seen in Figure 3.

More precisely, let

α2(x) =

{α21 0 ≤ x < xcα22 xc ≤ x < 2π

, (7)


0

30

200

400

6

|L(x

,)|

20

600

Constant coefficient

4

800

x

102

0 0

Fig. 2 Symbol L(x, ξ) of constant-coefficient operator L(x,D) = D2

0

30

1000

6

2000

|L(x

,)|

20

Piecewise constant coefficient

3000

4

x

102

0 0

Fig. 3 Symbol L(x, ξ) of operator L(x,D) = Dα2(x)D with piecewise constant coefficientα2(x) defined in (7) where α1 = 1, α2 = 2, and xc = 3π/4

where αk is the diffusivity and xc is the point of discontinuity in the coefficient.Based on Figure 3, if an eigenfunction is defined piecewise using sines andcosines that have periods 2π/ω1 on [0, xc) and 2π/ω2 on [xc, 2π), then bydifferentiation on each piece, we have the relation

α21ω

21 = α2

2ω22 . (8)


To see this, we note that if

A(x,D) = Dα2(x)D (9)

with α2(x) defined as in (7), then the symbol is given by

A(x, ω) =

{α21ω

2 0 ≤ x < xcα22ω

2 xc ≤ x < 2π(10)

and therefore a level curve C of A(x, ω) actually consist of two pieces,

C = {(x, ω1)|0 ≤ x < xc} ∪ {(x, ω2)|xc ≤ x < 2π} (11)

where we must have α21ω

21 = α2

2ω22 for A(x, ω) to remain constant on C. The

relation (8) can also be seen by an argument analogous to that presented in(Filoche et al., 2012) for the Dirichlet eigenvalue problem. An eigenfunction

Vj(x) =

{Vj1(x) 0 ≤ x < xcVj2(x) xc ≤ x < 2π

for which A(x,D)Vj(x) = λjVj(x) satisfies

−α21V′′j1(x) = λjVj1(x), 0 ≤ x < xc,

−α22V′′j1(x) = λjVj2(x), xc ≤ x < 2π,

and in view of the positivity of the eigenvalues of A(x,D), we therefore have

Vj1(x) = Aj1 cos

√λjx

α1+Bj1 sin

√λjx

α1, (12)

Vj2(x) = Aj2 cos

√λjx

α2+Bj2 sin

√λjx

α2. (13)

Setting ωjk =√λj/αk for k = 1, 2 yields (8).

While focusing on level curves indicates how the frequencies ω1 and ω2 fora particular eigenfunction relate to one another, it does not help us to estimateeither one. For this task, we apply the SAK principle, as previously appliedin (Lambers, 2012) to differential operators on a finite interval. In that paper,for A(x,D) chosen to be either a constant-coefficient and variable-coefficientoperator, the first several eigenvalues were computed, and then the areas ofthe subsets of phase space S(A, λj) were computed. It was observed that inmost cases, the area of the set difference S(A, λj)− S(A, λj−1) was very closeto 2π. It was therefore reasonable to conclude that the area of the region inphase space into which each eigenfunction is concentrated is approximately2π, whether the operator is constant-coefficient or variable-coefficient.

This is illustrated in Figure 4, in which we seek to estimate the frequenciesω11 and ω12 for the eigenfunction V1(x) corresponding to the smallest nonzeroeigenvalue λ1. We illustrate the process of estimating frequencies using anexample, in which the point of discontinuity is xc = 2πρ with ρ = 3

8 , and the


Fig. 4 The region in phase space in which the eigenfunction corresponding to the smallestnonzero eigenvalue is concentrated, for the case ρ = 3/8, α1 = 1 and α2 = 2.

diffusion coefficient is given by α1 = 1 and α2 = 2. Since we know that thearea of the region shown in Figure 4 is 2π, we then have

2π = ω113π

4+ ω12

(2π − 3π

4

). (14)

Based on the SAK principle that implies α1ω11 = α2ω12, this equation sim-plifies to

2π = 2ω123π

4+ ω12

5π

4, (15)

which has the solution

ω12 =8

11. (16)

Therefore, integer multiples of 811 are used as initial estimates of frequency val-

ues for the eigenfunctions. Figure 5 illustrates the accuracy of these estimates.

In general, our eigenvalue estimates λ(0)j , j = 1, 2, . . . , are given by

λ(0)j ≈ α

22ω

2j2, ωj2 =

k

φρ+ (1− ρ), k =

⌊j + 1

2

⌋, (17)

where bxc is the greatest integer less than or equal to x, and φ = α2/α1.This eigenvalue estimate can also be obtained by noting that α = α2/(φρ +(1 − ρ)) is the weighted harmonic mean of α1 and α2, with weights ρ and


Fig. 5 Smallest eigenvalues of A from (7), (9), with xc = 2πρ, ρ = 3/8, α1 = 1, α2 = 2(circles) compared to estimates obtained using (17) (crosses)

1− ρ, respectively. As in (Guidotti et al., 2006), if we consider the differentialoperator

A(x,D) = α2(x)D2

where α(x) varies smoothly between α1 and α2 in a neighborhood of xc, thenthe change of variable

y = Φ(x) =α

2π

ˆ x

0

1

α(ξ)dξ, α =

(1

2π

ˆ 2π

0

1

α(ξ)dξ

)−1yields the operator

A(y,D) = α2D2 − αα′(Φ−1(y))D.

It follows that as in (17), the eigenvalues are approximately equal to squaresof integer multiples of the harmonic average of α(x).

The initial guesses from (17) can serve as input to a root-finding method toobtain eigenvalues with high accuracy. Then, using these eigenvalues, unknownproperties of the corresponding eigenfunctions, including the amplitudes andphase shifts on each piece, must still be computed, and then the approxi-mate eigenfunctions can be constructed. For the spatial differential operator(∂/∂x)α2(x)(∂/∂x), the smallest eigenvalue is zero; therefore, the first eigen-function in the sequence is a constant function.


Since we are working with the interval [0, 2π], the piecewise eigenfunctionthat we are seeking will have the form similar to that suggested by (12), (13).Specifically, we prescribe

Vj(x) =

{Vj1(x) = Aj cos(ωj1x− θj) 0 ≤ x < 2πρVj2(x) = cos(ωj2x− τj) 2πρ ≤ x < 2π

, (18)

where 2πρ represents the discontinuity point, with 0 < ρ < 1. The parametersθj and τj represent that phase shifts for the two homogeneous materials; theparameters ωj2 and ωj1 represent the frequencies of the two homogeneousmaterials; the parameterAj represents the amplitude for the first homogeneousmaterial. For convenience, the eigenfunction will be normalized so that theamplitude for the second material is equal to one.

To obtain the eigenfunctions, all parameters except ωj2 are eliminatedusing the following conditions:

– 2π-periodicity,– continuity at the interface of the two homogeneous materials,– frequencies are related according to the SAK principle,– orthogonality with respect to the standard inner product, and– left- and right-hand co-normal derivatives must be equal at the interface

(see (Filoche et al., 2012) for discussion of this condition for the caseof Dirichlet boundary conditions; the same reasoning applies for periodicboundary conditions).

Once all parameters except ωj2 have been eliminated, a numerical method,the secant method (Burden and Faires, 2010), will be used to approximate thefrequency value ωj2, using (17) to obtain initial guesses. A numerical methodis necessary because eliminating the parameters results in a nonlinear equationfor ωj2.

The solution is to be represented as a linear combination using approximateeigenfunctions Vj(x) and eigenvalues λj for j = 1, 2, . . .. In the next section, itis shown that the eigenfunctions can be computed very accurately, with mostparameters determined analytically. Therefore, the solution can be representedusing the eigenfunction expansion

u(x, t) =

∞∑n=1

Vj(x)e−λjt〈Vj , f〉, (19)

where

〈f, g〉 =

ˆ 2π

0

f(x)g(x) dx (20)

is the standard inner product of real-valued functions on (0, 2π), and the initialcondition is u(x, 0) = f(x), where f(x) represents the initial temperatureprofile of the rod.

It is worth noting that the approach described in this section can beadapted to different boundary conditions, with the same initial guesses. To


illustrate this, we compute the smallest eigenvalues of the operator A ref-erenced in Figure 5, except with Dirichlet boundary conditions instead ofperiodic. Then, we compute the frequencies ωj2 for each eigenvalue λj , andcompare them to the initial guesses from (17). The comparison is shown inFigure 6. It can be seen that our proposed initial guesses are excellent approx-imations for alternating eigenvalues, with their midpoints serving as excellentinitial guesses for the remaining eigenvalues.

1 2 3 4 5 6 7 8 9 10 11

j

0

0.5

1

1.5

2

2.5

3

3.5

4

j2

Fig. 6 The circles are values of ωj2, j = 1, . . . , 10, corresponding to the 10 smallest eigen-values of the operator A from (7), (9), with xc = 2πρ, ρ = 3/8, α1 = 1, α2 = 2. The solidlines represent even integer multiples of 1

2ωj2 from (17). The dashed lines represent odd

integer multiples of 12ωj2.

4 Computing Eigenvalues and Eigenfunctions

In this section, we show how to compute, either analytically or numerically,the parameters involved in the description of the eigenfunctions (18).

4.1 Obtaining Parameter Values

The first parameter that will be eliminated is Aj . A formula for Aj cos(θ) andAj sin(θ) can be found using the conditions of 2π-periodicity and continuityat the interface. 2π-periodicity requires that the following relationship holds:

Vj(2π) = Vj(0). (21)


It follows that

Aj cos(θj) = cos(ωj22π − τj). (22)

Next, continuity at the interface implies

Aj cos(ωj12πρ− θj) = cos(wj22πρ− τj). (23)

Then, by applying a trigonometric identity to the left side of (23) and substi-tuting equation (22) into (23), we obtain

cos(ωj22π− τj) cos(ωj12πρ) +Aj sin(ωj12πρ) sin(θj) = cos(wj22πρ− τj) (24)

and, thus,

Aj sin(θj) =cos(wj22πρ− τj)− cos(ωj12πρ) cos(ωj22π − τj)

sin(ωj12πρ), (25)

assuming that sin(ωj12πρ) 6= 0. This special case will be discussed later in thissection.

By obtaining the expressions (22) and (25) for Aj cos(θj) and Aj sin(θj),respectively, the parameters θj and Aj can be eliminated. After substitutingthese expressions into (18), the two-material general eigenfunction has theform

Vj(x) =

cos(ωj22π − τj) cos(ωj1x)+cos(ωj22πρ−τj)−cos(ωj12πρ) cos(ωj22π−τj)

sin(ωj12πρ)sin(ωj1x) 0 ≤ x < 2πρ

cos(ωj2x− τj) 2πρ ≤ x < 2π(26)

The next parameter that will be eliminated is ωj1, using (8). As before, letφ = α2

α1so that ωj1 = φωj2. Substituting in the equation for ωj1, the equation

for our eigenfunction is now in this form:

Vj(x) =

cos(ωj22π − τj) cos(φωj2x)+cos(ωj22πρ−τj)−cos(φωj22πρ) cos(ωj22π−τj)

sin(φωj22πρ)sin(φωj2x) 0 ≤ x < 2πρ

cos(ωj2x− τj) 2πρ ≤ x < 2π

.

(27)Next, τj is eliminated by using the condition that the left- and right-hand

co-normal derivatives must be equal, and the requirement of orthogonality.The matching co-normal derivative condition specifies that the eigenfunctionmust satisfy the following equation:

α21V′j1(2πρ) = α2

2V′j2(2πρ). (28)

This condition comes from the weak form of the eigenvalue problem, adaptedfrom (Filoche et al., 2012), in which we seek λ ∈ R and V ∈ H1

per((0, 2π))(that is, 2π-periodic functions in H1((0, 2π))) such that

ˆ 2π

0

α2(x)V ′(x)ϕ′(x) dx = λ

ˆ 2π

0

V (x)ϕ(x) dx,


for all ϕ ∈ C∞per((0, 2π)), the space of 2π-periodic functions in C∞((0, 2π)).Applying the condition (28) to (27), and using φ = α2

α1. yields

−φ sin(ωj22πρ− τj) = − cos(ωj22π − τj) sin(φωj22πρ) +

cos(ωj22πρ− τj)− cos(φωj22πρ) cos(ωj22π − τj)sin(φωj22πρ)

×

cos(φωj22πρ). (29)

Next, the trigonometric functions should be expanded using trigonometricidentities so that τj can be isolated.

−φ sin(ωj22πρ) cos(τj) + φ cos(ωj22πρ) sin(τj)

= − sin(φωj22πρ)[cos(ωj22π) cos(τj) + sin(ωj22π) sin(τj)] +

cos(φωj22πρ)

sin(φωj22πρ)[cos(ωj22πρ) cos(τj) + sin(ωj22πρ) sin(τj)]−

cos2(φωj22πρ)

sin(φωj22πρ)[cos(ωj22π) cos(τj) + sin(ωj22π) sin(τj)] (30)

We see that either cos(τj) or sin(τj) can be factored out of each term of thisequation:

0 = cos(τj)[− cos(ωj22π) + cos(φωj22πρ) cos(ωj22πρ) +

φ sin(ωj22πρ) sin(φωj22πρ)] +

sin(τj)[− sin(ωj22π) + cos(φωj22πρ) sin(ωj22πρ)−φ cos(ωj22πρ) sin(φωj22πρ)] (31)

Finally, the eigenfunction needs satisfy the orthogonality condition

〈Vj , V0〉 =

ˆ 2π

0

Vj(x)V0(x) dx = 0, (32)

where V0 is the eigenfunction that corresponds to the smallest eigenvalueλ0 = 0. Because V0(x) is a constant function, the equation from the orthog-

onality condition reduces to´ 2π0Vj(x) dx = 0. Because the eigenfunction is a

piecewise-defined eigenfunction with the discontinuity point 2πρ, the orthog-onality condition can be written as

ˆ 2πρ

0

Vj1(x) dx+

ˆ 2π

2πρ

Vj2(x) dx = 0 (33)

which is equivalent toˆ 2πρ

0

cos(ωj22π − τj) cos(φωj2x)+

cos(ωj22πρ− τj)− cos(φωj22πρ) cos(ωj22π − τj)sin(φωj22πρ)

sin(φωj2x) dx+

ˆ 2π

2πρ

cos(ωj2x− τj) dx = 0.(34)


Then, the integral can be evaluated as follows:

0 = cos(ωj22π − τj) sin2(φωj22πρ)− cos(ωj22πρ− τj) cos(φωj22πρ) +

cos(ωj22π − τj) cos2(φωj22πρ) + cos(ωj22πρ− τj)−cos(φωj22πρ) cos(ωj22π − τj) + φ sin(φωj22πρ)[sin(ωj22π) cos(τj)−cos(ωj22π) sin(τj)− sin(ωj22πρ) cos(τj) + cos(ωj22πρ) sin(τj)]. (35)

Next, the evaluated integral should be simplified and the cos(τj) and sin(τj)factors should be grouped together.

0 = cos(τj)[cos(ωj22π)− cos(ωj22πρ) cos(φωj22πρ) + cos(ωj22πρ)−cos(φωj22πρ) cos(ωj22πρ) + φ sin(ωj22πρ)(sin(ωj22π)−sin(ωj22πρ))]

+ sin(τj)[sin(ωj22π)− sin(ωj22πρ) cos(φωj22πρ) + sin(ωj22πρ)−cos(φωj22πρ) sin(ωj22πρ)− φ sin(ωj22πρ)(cos(ωj22π)−cos(ωj22πρ))]. (36)

Now, τj can be eliminated by taking equation (31) and (36) and makingthe following homogeneous system of equations with the unknowns, cos(τj)and sin(τj): [

G1 G2

H1 H2

] [cos(τj)sin(τj)

]=

[00

](37)

The following four expressions make up the homogenous system:

G1 = cos(ωj22π)− cos(ωj22πρ) cos(φωj22πρ) + cos(ωj22πρ)−cos(φωj22πρ) cos(ωj22πρ) + φ sin(ωj22πρ)(sin(ωj22π)−sin(ωj22πρ)) (38)

G2 = sin(ωj22π)− sin(ωj22πρ) cos(φωj22πρ) + sin(ωj22πρ)−cos(φωj22πρ) sin(ωj22πρ)− φ sin(ωj22πρ)(cos(ωj22π)−cos(ωj22πρ)) (39)

H1 = − cos(ωj22π) + cos(φωj22πρ) cos(ωj22πρ) + φ sin(ωj22πρ) sin(φωj22πρ)(40)

H2 = − sin(ωj22π) + cos(φωj22πρ) sin(ωj22πρ)− φ cos(ωj22πρ) sin(φωj22πρ),(41)

where G1 and G2 are coefficients of cos(τj) and sin(τj), respectively, from theorthogonality condition, and H1 and H2 are the coefficients from the matchingof co-normal derivatives.

In order for this system to have a nontrivial solution, the determinantof the matrix must be zero; therefore, the equation G1H2 − G2H1 = 0 is


solved iteratively using the secant method to obtain ωj2, or, equivalently, theeigenvalue λj = α2

2ω2j2. The secant method is used because it has a rapid rate

of convergence (Burden and Faires, 2010), but unlike Newton’s method, whichconverges more rapidly, it does not require evaluating a derivative, which wouldbe very complex for this determinant.

The secant method requires two initial guesses λ(0)j and λ

(1)j . In our im-

plementation, we set λ(0)j based on (17), and then perturb λ

(0)j to obtain the

second initial guess λ(1)j . The perturbation is made so as to be closer to a root;

that is, if the objective function G1H2 − G2H1 is positive and increasing at

λ(0)j , then we perturb so that λ

(1)j < λ

(0)j .

Once λ(k)j , k = 0, 1, 2, . . . , converges to λ∗j , we use the observation that, as

shown in Figure 5, nonzero eigenvalues are clustered in pairs, and the eigen-value estimate (17) falls between the two nearest eigenvalues in each pair. We

set λ(0)j+1 = 2λ

(0)j −λ∗j , to obtain a more accurate initial guess than (17) would

provide. Then, λ(0)j+1 is perturbed in a similar manner as λ

(0)j was to obtain

λ(1)j+1.

4.2 Excluding False Eigenvalues

We have previously assumed that sin(ωj12πρ) 6= 0 to obtain an expressionfor Aj sin(θj); however, we must also investigate whether it is still possible toobtain eigenfunctions when sin(ωj12πρ) = 0. Therefore, we assume ωj1 = k

2ρ ,where k is an integer. Then, this expression for ωj1 must be substituted intoVj(x) and then applied to the conditions for periodicity, continuity at theinterface, orthogonality, relation of frequencies based on the SAK principle,and matching of co-normal derivatives at the interface. We begin with

Vj(x) =

{Vj1 = Aj cos( k2ρx− θj) 0 ≤ x < 2πρ

Vj2 = cos( kφ2ρx− τj) 2πρ ≤ x < 2π.

(42)

After the substitution of ωj1, the periodicity condition is as follows:

Aj cos(θj) = cos

(k

φ2ρ2π − τj

). (43)

The continuity condition yields

(−1)k cos

(kπ

φρ− τj

)= cos

(kπ

φ− τj

)(44)

Then, applying the orthogonality condition (33), we obtain

2Ajρ

k[sin(kπ − θj) + sin(θj)] +

2φρ

k

[sin

(kπ

φρ− τj

)− sin

(kπ

φ− τj

)]= 0.

(45)


Equation (45) can be solved for Aj sin(θj), to obtain

Aj sin(θj) =φ sin(kπφρ − τj)− φ sin(kπφ − τj)

1− (−1)k, (46)

provided that k is odd. The case where k is even will be considered at the endof this discussion.

By enforcing equality of co-normal derivatives from (28), we obtain

−Ajk

2ρα21 sin (kπ − θj) = − k

2ρα22 sin

(kπ

φ− τj

). (47)

Next, we rearrange this equation to obtain another expression for Aj sin(θj):

Aj sin(θj) = −(−1)kφ sin

(kπ

φ− τj

)(48)

Then, using equation (46) and equation (48) we obtain

(−1)k sin

(kπ

φρ− τj

)= sin

(kπ

φ− τj

). (49)

Equation (44) from the continuity condition and equation (49) can be dividedto obtain a condition on k:

tan

(kπ

φ− τj

)= tan

(kπ

φρ− τj

). (50)

We then use the π-periodicity of tan to obtain the condition

kπ

φ=kπ

φρ+ nπ, (51)

or, equivalently,

k − nφ =k

ρ, (52)

where n is an integer. It is helpful to rearrange this equation to isolate ρ andn:

ρ =k

k − nφ(53)

n =k

φ

(1− 1

ρ

)(54)

Because ρmust be on the interval (0, 1), we know that n � 0 because otherwise,we would have ρ < 0 or ρ ≥ 1. Therefore, n < 0 is a necessary condition foran eigenfunction to exist.

For a given integer k, we can test whether k/φ(1−1/ρ) is an integer, and ifso, then α2

1ω2j1 is an eigenvalue. If k is even, we obtain from (45) the condition

kπ

φ=kπ

φρ+ 2nπ, (55)

which can be treated similarly.


4.3 Construction of Eigenfunctions

We now summarize the construction of approximate eigenfunctions. For j =1, 2, . . ., we proceed as follows:

1. Construct the initial guesses for ωj2 using (17) if j is odd, or using λ(0)j =

2λ(0)j−1 − λ∗j−1 if j is even, where λ∗j−1 is the (previously) computed value

of λj−1.2. Then, the ωj2 frequency value can be found by solving the equation G1H2−G2H1 = 0 for ωj2, where G1, G2, H1, H2 are defined as in (38)-(41).

3. Set ωj1 = α2

α1ωj2. If ωj1 is of the form k

2ρ , where k is an integer, then checkwhether it corresponds to a false eigenvalue, as discussed in Section 4.2,using (51) if k is odd, or (55) if k is even.

4. Using the homogeneous system (37), cos(τj) and sin(τj) can be computedas follows:

(cos(τj), sin(τj)) =

(−H2√H2

1 +H22

,H1√

H21 +H2

2

). (56)

5. Compute the amplitude Aj of the left piece Vj1(x) as follows:

Aj =

√(cos(ωj22π − τj))2 +

(cos(ωj22πρ− τj)− cos(ωj12πρ) cos(ωj22π − τj)

sin(ωj12πρ)

)2

,

(57)6. Obtain θj from

tan(θj) =cos(wj22πρ− τj)− cos(ωj12πρ) cos(ωj22π − τj)

sin(ωj12πρ) cos(ωj22π − τj). (58)

It is worth noting that from (37), cos(τj) and sin(τj) can just as easily beobtained from

(cos(τj), sin(τj)) =

(−G2√G2

1 +G22

,G1√

G21 +G2

2

). (59)

However, we have concluded by experimentation that it is significantly morenumerically stable to use H1 and H2, because for half of the eigenvalues,G1, G2 ≈ 0, and thus the values of cos(τj) and sin(τj) are contaminated byroundoff error.

An important property of the above algorithm is that each pair of nonzeroeigenvalues is computed independently of other pairs. It follows that the pro-cess of constructing a basis of approximate eigenfunctions is highly paralleliz-able.


5 Solving PDE

Once the approximate eigenfunctions are constructed, the PDE can be solved.Modeling the 2π-periodic heat equation gives the following PDE:

∂u

∂t=

∂

∂x

(α2(x)

∂u

∂x

), 0 < x < 2π (60)

whereu(x, 0) = f(x). (61)

This initial-boundary value problem has a solution of the form:

u(x, t) =

∞∑j=0

ujeλjtVj(x), (62)

whereλj = −α2

2ω2j2 (63)

and

uj =〈Vj , f〉〈Vj , Vj〉

=

´ 2π0Vj(x)f(x) dx´ 2π

0Vj(x)2 dx

. (64)

The denominator can be evaluated analytically:

ˆ 2π

0

Vj(x)2 dx =

ˆ 2πρ

0

A2j cos2(ωj1x− θj)dx+

ˆ 2π

2πρ

cos2(ωj2x− τj) dx

= A2j

ˆ 2πρ

0

1

2(1 + cos(2ωj1x− 2θj)) dx

+

ˆ 2π

2πρ

1

2(1 + cos(2ωj2x− 2τj)) dx

=A2j

2

[x+

sin(2ωj1x− 2θj)

2ωj1

]∣∣∣∣2πρ0

+

[x

2+

sin(2ωj2x− 2τj)

4ωj2

]∣∣∣∣2π2πρ

=A2j

2

([2πρ+

sin(4ωj1πρ− 2θj)

2ωj1

]−[

sin(−2θj)

2ωj1

])+

([π +

sin(4ωj2π − 2τj)

4ωj2

]−[πρ+

sin(4ωj2πρ− 2τj)

2ωj2

]).

(65)

The procedure for solving (60), (61) at time t = T is as follows:

1. Determine a positive integer J such that e−λ(0)2J T < ε, where λ

(0)2j , for any

positive integer j, is defined in (17) and ε is an absolute error tolerance.2. Construct the eigenfunctions Vj(x), j = 1, 2, . . . , J as in Section 4.3.3. Compute the inner product 〈Vj , Vj〉 using (65).4. Use a quadrature rule to obtain the numerators 〈Vj , f〉 of (64) for j =

0, 1, . . . , 2J .


5. Compute uj , j = 0, 1, . . . , 2J using (64).6. Compute u(x, T ) using (62).

It should be noted that the inner products 〈Vj , f〉 from (64) used in (62) can beobtained either by evaluating the integrals in (64) directly, or using the Fourierseries coefficients of f(x), as the inner products 〈Vj , cos kx〉 and 〈Vj , sin kx〉can be evaluated analytically.

6 Numerical Results

6.1 Finding Eigenfunctions

The following tables list the numerical results from approximating the ωj2frequency values numerically with the secant method, using the methods de-scribed in Section 4. The tables report the eigenvalues computed in this man-ner, as well as the approximate eigenvalues obtained by computing the eigen-values of a matrix that is a finite difference representation of the differentialoperator

L = −(∂/∂x)α(x)2(∂/∂x), (66)

where α2(x) is defined in (7) with xc = 2πρ, using the Matlab function eigs.The errors listed in the third column are the absolute differences betweencorresponding approximate eigenvalues. The errors listed in the fifth columnare the absolute differences between the last two iterates computed by thesecant method to obtain ωj2, then multiplied by 2α2

2ωj2 to account for thepropagation of error by the formula λj = (α2ωj2)2. The iteration count listedis the number of iterations needed by the secant method to compute eachfrequency value with a tolerance of 10−12.

In the first trial, α1 = 1, α2 = 2, and ρ = 38 . The results are shown in

Table 1. In the second trial, we consider the case where the ρ is irrational, to

j λj (FD) error (FD) λj (SAK) error (SAK) iterations0 0.00000 1.9073e-06 0.00000 0.0000e+00 N/A1 0.70612 1.9065e-06 0.70612 1.0543e-12 42 0.74970 1.1729e-06 0.74970 4.5945e-14 133 1.41365 1.5047e-05 1.41366 0.0000e+00 44 1.49759 9.7335e-06 1.49760 2.6603e-15 115 2.12401 4.9602e-05 2.12406 5.3050e-12 46 2.24195 3.4769e-05 2.24199 5.9739e-13 97 2.83862 1.1357e-04 2.83873 0.0000e+00 78 2.98124 8.8403e-05 2.98133 2.9657e-13 59 3.55870 2.1187e-04 3.55892 1.2644e-14 8

10 3.71434 1.8605e-04 3.71452 0.0000e+00 5

Table 1 Eigenvalues computed using eig, and using the method from Section 4, withα1 = 1, α2 = 2, and ρ = 3/8

prevent the discontinuity from coinciding with a grid point. We set α1 = 1,


α2 = 2, and ρ =√2

2π . The results are shown in Table 2. In the third trial, we

j λj (FD) error (FD) λj (SAK) error (SAK) iterations0 0.00000 2.6974e-06 0.00000 0.0000e+00 N/A1 0.75545 6.9844e-05 0.75538 6.7091e-16 52 0.88567 5.8869e-04 0.88508 6.1183e-12 93 1.54483 5.8179e-04 1.54425 0.0000e+00 94 1.71982 5.0918e-04 1.71931 8.8569e-14 55 2.39425 1.7023e-03 2.39255 8.5001e-15 66 2.49764 9.9473e-05 2.49754 0.0000e+00 117 3.24999 7.6655e-05 3.25006 1.1547e-14 78 3.28558 2.5892e-03 3.28299 3.4991e-14 79 4.01042 3.8098e-04 4.01004 0.0000e+00 5

10 4.16204 2.1999e-03 4.15984 4.8179e-12 11

Table 2 Eigenvalues computed using eig, and using the method from Section 4, withα1 = 1, α2 = 2, and ρ =

√2/(2π)

test the method with a wider gap between coefficient values, setting α1 = 1,α2 = 4, ρ = 3

8 . The results are shown in Table 3. In the fourth trial, we test

j λj (FD) error (FD) λj (SAK) error (SAK) iterations0 0.00000 0.0000e+00 0.00000 0.0000e+00 N/A1 0.38777 1.6891e-06 0.38777 1.1012e-11 72 0.56697 1.9321e-06 0.56697 7.3884e-12 53 0.88006 1.0664e-05 0.88007 5.9031e-12 84 0.98041 2.0143e-05 0.98043 2.3686e-13 65 1.36014 5.9469e-05 1.36019 1.0631e-13 116 1.48804 3.6118e-05 1.48808 1.0573e-14 77 1.78815 1.0899e-04 1.78826 1.2706e-14 68 1.95589 1.5238e-04 1.95604 4.1695e-14 89 2.33935 2.9489e-04 2.33965 3.3248e-14 9

10 2.37468 1.3570e-04 2.37481 3.3748e-14 8


the method with swapped coefficient values, setting α1 = 4, α2 = 1, ρ = 38 .

The results are shown in Table 4.We also examine the behavior of the error in the approximate eigenvalues

as the size of the matrix increases. For the case of ρ = 3/8, α1 = 1 and α2 = 2,we compare the eigenvalue λ1 computed using the method of Section 4 againstthat computed using eigs for a matrix of size N×N , with N = 256, 512, 1024.The results are shown in Table 5. It can be seen that as N increases, theeigenvalues computed from the finite difference representation of the operatorL converges quadratically to the eigenvalue computed using the method ofSection 4, which provides further numerical evidence that the latter eigenvalueis the exact eigenvalue, up to the error tolerance used in the secant method.


j λj (FD) error (FD) λj (SAK) error (SAK) iterations0 0.00000 2.6974e-06 0.00000 0.0000e+00 N/A1 1.12098 2.9089e-06 1.12098 4.9782e-16 72 1.53831 4.8667e-06 1.53832 1.3663e-15 93 2.53441 2.7231e-05 2.53444 5.6051e-13 84 3.04544 3.5498e-05 3.04547 5.2205e-13 45 4.05027 1.0369e-04 4.05037 4.8853e-12 96 4.42367 9.0057e-05 4.42376 7.8582e-15 77 5.50057 1.7953e-04 5.50075 8.6964e-13 88 5.59008 2.6345e-04 5.59034 2.7309e-12 79 6.67457 4.3512e-04 6.67500 0.0000e+00 7

10 7.12572 5.2723e-04 7.12625 2.5318e-14 9


N λ1 (SAK) λ1 eig error256 0.70612 0.70609 3.0505e-05512 0.70612 0.70611 7.6261e-06

1024 0.70612 0.70612 1.9065e-06

Table 5 Convergence of eigenvalues from finite difference representation to eigenvaluescomputed using secant method with initial guess from SAK principle

The quadratic convergence to the exact eigenvalue is expected, as the matrixis constructed using a centered difference approximation, which is second-order accurate. The same convergence behavior is observed for the ten smallesteigenvalues.

6.2 Results from Solving a PDE

We now apply the approach of Sections 4 and 5 to compute the solution ofthe diffusion problem ut + Lu = 0 on (0, 2π), where L is defined in (66), withperiodic boundary conditions and initial condition u(x, 0) = f(x). For thefollowing experiments, we use ρ = 3/8, α1 = 1 and α2 = 2.

In the first experiment, the initial data is the periodic function u(x, 0) =sin(2x). The results are shown in Figure 7. As the final time is increased, thesolution behaves as expected and becomes more smooth. Figure 8 illustratesthe solution computed using initial data being a characteristic function

u(x, 0) =

{1, π/4 < x < 3π/40, otherwise

(67)

As time increases, the solution also behaves as expected and becomes moresmooth over time. For this problem, we compare the efficiency of our approachto a standard method, Crank-Nicholson, which is second-order accurate andunconditionally stable (Gustafsson et al., 1995). Although the time step canbe chosen to be large while still ensuring stability, if it is chosen too large,


Fig. 7 Solution with α1 = 1, α2 = 2, ρ = 3/8 and u(x, 0) = sin(2x)

then the solution will exhibit high-frequency oscillations due to the Gibbs’phenomenon. In Table 6, we list the computational time needed to compute asolution using Crank-Nicholson that does not have high-frequency oscillationat the final time, as well as the time needed to compute a solution using theMatlab solver ode15s (Shampine and Reichelt, 1997), and compare to thecomputational time needed to compute the solution at the same final timeusing the methods of Section 4 and 5 to compute an eigenfunction expansion,using the approximate eigenfunctions computed as described in these sections.

In each experiment, the expansion is truncated after m terms, where m ischosen based on checking e−λjT against a threshold; 10−3 was used in theseexperiments, the same default relative tolerance used by ode15s. It followsthat the larger the value of T , the fewer terms are needed in the approximateeigenfunction expansion. As expected, it can be observed that as the finaltime T increases, using an eigenfunction expansion requires fewer terms, dueto the exponential decay of the coefficients in the expansion. Furthermore, asT increases, the eigenfunction expansion yields the solution in far less com-putational time, due to the need for fewer terms while Crank-Nicholson andode15s require more time steps.

Finally, it should be noted that as the number of grid points in the spatialdiscretization,N , doubles, the time required by Crank-Nicholson increases by afactor of roughly 4, due the need to take more time steps to maintain accuracy,and the time required by ode15s approximately doubles. The eigenfunctionexpansion, however, incurs a much smaller increase, that is actually negligiblefor larger values of T for which fewer terms are needed. This is because most


Fig. 8 Solution with α1 = 1, α2 = 2, ρ = 3/8 and u(x, 0) is a characteristic function

of the computation is independent of N ; the only exception is the computationof the coefficients of the initial data in the basis of eigenfunctions.

N = 1024 grid pointsCrank-Nicholson ode15s eig. exp.

T time ∆t time avg. ∆t time m0.01 0.043 1.25e-04 0.082 9.0e-5 0.0086 370.1 0.100 5.00e-04 0.092 7.2e-4 0.0037 11

1 0.390 1.25e-03 0.101 6.1e-3 0.0012 3N = 2048 grid points

Crank-Nicholson ode15s eig. exp.T time ∆t time avg. ∆t time m

0.01 0.159 6.25e-05 0.153 7.8e-5 0.0105 370.1 0.388 2.50e-04 0.185 6.4e-4 0.0040 11

1 1.493 1.25e-04 0.220 5.5e-3 0.0012 3

Table 6 Computational time, in seconds, using Crank-Nicholson with time step ∆t, ode15swith average time step ∆t, and eigenfunction expansion with m terms, to solve ut +Lu = 0with L defined in (66), α1 = 1, α2 = 2, ρ = 3/8, and initial data from (67).

7 Conclusions

The outcome of this project was overall successful. An accurate, efficient, andparallelizable algorithm for computing eigenfunctions of differential operators


was used to model the heat diffusion through two homogeneous materials wasdesigned and implemented. This method substantially simplifies the solutionof such models with discontinuous coefficients and bypasses the limitations ofexisting numerical methods; however, there is future work that can be donewith this project to improve and expand it.

By solving problems with discontinuous coefficients accurately and effi-ciently, the method can substantially aid researchers in science and engineeringwho need to simulate such heat diffusion processes. To improve this method,a more reliable way to find the zeros of the determinant should be explored.Alternatively, working with the smallest singular value of the matrix shouldbe investigated, especially for generalization to problems featuring more thantwo materials, because it is a more reliable indicator of distance to the nearestnon-invertible matrix than the determinant is (Golub and van Loan, 1996).

For continuation of this work, the case of more than two homogeneous ma-terials is being investigated, as similar conditions apply to the eigenfunctionsat the boundaries and interfaces. Due to the structure implied by the resultingequations, frequencies can still be obtained by computing the roots of the de-terminant of a 2×2 matrix, regardless of the number of materials (Long et al.,2017). This work can not only improve insight into the qualitative behav-ior of solutions of different types of equations with discontinuous coefficients,but also lead to new methods for computing eigenfunctions of operators withsmoothly varying coefficients. This project is also being continued by studyingthe behavior of these equations in two-dimensional or three-dimensional mod-els, as the SAK principle is not limited to one space dimension. Current workon the relatively simple case of a single interface within a two-dimensionaldomain has yielded similarly accurate results (Aurko and Lambers, 2017).

Acknowledgments

The first author was partially supported by a grant from the Center for Un-dergraduate Research at the University of Southern Mississippi.

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