exponential time differencing crank–nicolson method with a quartic spline approximation for...

A Fully Adaptive Approximation forQuenching-Type Reaction-Diffusion Equationsover Circular DomainsMatthew A. Beauregard,1 Qin Sheng2

1Department of Mathematics, Clarkson University, Potsdam, New York 13699-58152Department of Mathematics and Center for Astrophysics, Space Physics andEngineering Research, Baylor University, One Bear Place, Waco, Texas 76798-7328

Received 13 February 2013; revised 28 August 2013; accepted 19 September 2013Published online 26 October 2013 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/num.21820

This article studies a fully adaptive finite difference method for solving quenching-type nonlinear reaction-diffusion equations over circular domains. Although an auxiliary condition at the origin and radial symmetryare imposed, adaptations are accomplished via arc-length-based monitoring functions in space and time,respectively. The monotonicity and positivity of the numerical solution are proved following a suitable gridconstraint, and the numerical stability is ensured in the von Neumann sense. Theoretical bounds of thecritical quenching radius are obtained and then refined through the computation. Computational examplesare provided to illustrate the effectiveness and plausibility of the new adaptive computational proceduredeveloped. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 472–489, 2014

Keywords: quenching phenomenon; reaction-diffusion equation; circular domain; grid adaptation; polartransformation

I. INTRODUCTION

Applications of reaction-diffusion differential equations are ubiquitous in nature, especially inmodeling phenomena appearing in biology, chemistry, and engineering fields. These applicationshave been the motivation behind numerous theoretical and computational explorations in modernmathematics, subsequently providing intricate and in-depth insights into the scientific phenomenatargeted. A particularly interesting case is when rapid changes in a physical system, such as thetemperature in a combustion chamber, are to be modeled. In this situation, strong singularitiesmay occur in solutions of the equations formulated. This may complicate the development of themathematical theory as well as its numerical applications.

Correspondence to: Matthew A. Beauregard, Department of Mathematics, Clarkson University, Potsdam, New York13699-5815 (e-mail: [email protected])Contract grant sponsor: Air Force Research Laboratory (Q.S.); contract grant number: F-5400-04-06-SC01-00Contract grant sponsor: Baylor University (Q.S.); contract grant number: URC-18–330107-12

© 2013 Wiley Periodicals, Inc.

QUENCHING-TYPE REACTION-DIFFUSION EQUATIONS 473

Consider a solid fuel ignition model that approximates the temperature distribution through-out the combustion chamber asymptotically close to ignition [1, 2]. Although the temperaturemonotonically increases toward a finite critical combustion temperature, the rate of change ofthe temperature dramatically increases and blows up. Equivalently, although the solution of themodeling differential equation remains bounded throughout the process, its temporal derivativedevelops a strong singularity and quickly becomes unbounded. This phenomenon is referred toas quenching, and the time for which quenching occurs is called the quenching time [3–5]. Thedevelopment of quenching singularities presents unique and interesting mathematical challenges.

For a fixed combustor geometry, it is known that there exists a critical size for which quenchingcan be observed [4]. Determining or showing the existence of critical criteria for which quenchingmust occur has persisted as an important mathematical inquiry analytically and computation-ally [4, 6–8]. As the quenching time is approached, the speed at which a quenching singularityforms is astounding. Thus, novel and new algorithms are fundamental to ensure the computa-tional efficiency and effectiveness [2, 7, 9]. Methods that incorporate temporal adaptations havebeen developed. These methods have seen both successes and limitations (see [5] and referencestherein). Recently, various spatial adaptations have been introduced for quenching simulations[6, 10]. However, the most appropriate way to generate new spatial grids is unclear. The numericalanalysis of full adaptations, that is, combined temporal and spatial adaptations, is in its infancy.Nevertheless, attempts to modify traditional adaptation principles have been made and show sub-stantial advantages in grid constructions [11]. Hence, the development of exponentially evolvinggrids (EEG) is an alternative grid adaptation procedure and numerical experiments have shownpotential [6].

On the other hand, approximations of quenching solutions defined over more realistic spa-tial domains remain of primary importance. This article aims at circular geometries in space,while highlighting advantages of a fully adaptive method employing EEG. Colloquially, this pro-vides additional validity and confidence in the application of adaptive computations that utilizesmodified adaptation principles as introduced in [11] while increasing the domain sophistication.

This article is organized as follows. The next section studies a standard model reaction-diffusionproblem commonly associated to solid fuel combustion [1, 4]. An auxiliary condition is derivedat the origin based on l’Hôpital’s rule. Theoretical bounds of the critical quenching domain radiusare estimated. These results are later verified through computations via the newly developednumerical method. Section 3 details the fully adaptive scheme that approximates the nonlinearequations on circular domains. A rigorous numerical analysis of the difference scheme can befound in Section 4. The physical model requires the solution to be positive and monotonicallyincreasing. It is proven that this is guaranteed if a key constraint on the mesh step sizes is satisfied.Further, it is shown that the numerical method is linearly stable without further constraint. Severalsimulation experiments are presented in Section 5. They include a computational verification ofthe early prediction of the critical quenching domain radius, an exploration of the spatial gridevolution, and an investigation of the effect of a nonlinear memory term [12]. Finally, conclusionsand brief remarks are documented in Section 6.

II. QUENCHING-TYPE REACTION-DIFFUSION EQUATIONS

Let � = {(r , θ) |0 ≤ r < �, θ ∈ [0, 2π)} and ∂� be its boundary. Consider a solid fuel combus-tion scenario where the activation energy has decoupled from the mass-species fraction [1]. Inthis case, the heat distribution u close to activation or ignition can be ideally approximated by thesingular nonlinear reaction-diffusion equation

Numerical Methods for Partial Differential Equations DOI 10.1002/num

474 BEAUREGARD AND SHENG

ut = �u + f (u) , (r , θ) ∈ �, {0} , t > 0, (2.1)

u (�, θ , t) = 0, θ ∈ [0, 2π) , t > 0, (2.2)

u (r , θ , 0) = u0 (r , θ) � 1, (r , θ) ∈ �, (2.3)

where �u = 1r(rur)r + 1

r2 uθθ and f (u) is a singular absorption term that is strictly increasing foru ∈ [0, 1) such that

f (0) = f0 > 0, limu→1−f (u) = ∞.

A sufficient condition for quenching to occur is u → 1− as t → T − < ∞ in � ⊆ � [13]. Insuch cases, the solution u is said to quench at the quenching time T over the domain �. The set� is called the quenching location set. Given the circular geometry, it is known that there existsa critical size of the radius, �∗ > 0, for which quenching singularities persist if � ≥ �∗ [4].

The initial function, u0 (r , θ) , is assumed to be radially symmetric. This implies that the solu-tion of (2.1)–(2.3) is radially symmetric, hence the θ -dependence can be dropped. It follows thatthe quenching location is known to be at the origin, r = 0. Naturally, in contrast to that of rec-tangular domains, the initial-boundary value problem (2.1)–(2.3) is not defined at the origin asa result of the polar coordinate transformation. This drawback can be eased by introducing anauxiliary condition based on a limit feature of the Eq. (2.1) as r → 0+. To this end, l’Hôpital’srule is applied to the diffusion term. This leads to the following time-dependent constraint,

ut (0, t) = 2urr (0, t) + f (u (0, t)) .

Thus, on a rescaling, Eqs. (2.1)–(2.3) can be reformulated as

ut = α

r(rur)r + f (u) , 0 < r < 1, t > 0, (2.4)

ut (0, t) = 2αurr (0, t) + f (u (0, t)) , t > 0, (2.5)

u (1, t) = 0, t > 0, (2.6)

u (r , 0) = u0 (r) , 0 ≤ r < 1, (2.7)

where α = �−2.Although much of the recent work has focused on the existence of a critical quenching domain

and quenching times [4, 8], determining theoretical bounds or estimates to the critical quenchingdomain remains a difficult task, as both the shape and size influence their consideration [5]. Con-sequently, various approximations are utilized to provide numerical estimates to actual criticalquenching domains [3, 10].

The radial symmetry of (2.4)–(2.7) can be exploited to yield a theoretical critical bound �∗ for atypical absorption function f (u) = 1/ (1 − u). Our theoretical estimates rely on the constructionof upper and lower solutions bounding the critical α∗ value following ideas proposed in [14]. Thisprocedure is intimately linked to the utilization of Nagumo’s Lemma, that is,

Lemma 2.1 ([14]). Ifvt ≤ α

r(rvr)r + f (v) , 0 < r < 1, t > 0,

vt (0, t) ≤ 2αvrr (0, t) + f (v (0, t)) , v (1) ≤ 0, t > 0,

v ≤ 0, 0 ≤ r < 1, t = 0,



then v ≤ u for 0 ≤ r ≤ 1 and t > 0, where u is the solution of the initial-boundary valueproblem (2.4)–(2.7). Similarly, for the same u if

wt ≥ α

r(rwr)r + f (w) , 0 < r < 1, t > 0,

wt (0, t) ≥ 2αwrr (0, t) + f (w (0, t)) , w (1) ≥ 0, t > 0,

w ≥ 0, 0 ≤ r < 1, t = 0,

then w ≥ u for 0 ≤ r ≤ 1 and t > 0.If the function w (r , t) is bounded below unity for a given α then u (r , t) will not quench. To

obtain an upper solution, w (r , t) , amounts to solving a steady-state problem, namely,

α

(wrr + 1

rwr

)+ f (w) = 0, 0 < r < 1,

2αwrr (0) + f (w (0)) = 0, w (1) = 0.

The value α∗ for which the maximum of 0 ≤ w (r) < 1 is adequately close to unity is determinedthrough constructing a sequence of maximum values from its numerical solutions for descendingα values. If the maximum value of w(r) is greater than one, then the calculation must be repeatedstarting from a decreased value of α. Following this process, we found that w(r) is bounded belowunity when α > 0.76664, or � < 1.1421. Therefore, the solution u (r , t) of (2.4)–(2.7) cannotquench and is defined for all t for circular domains with radii less than � < 1.1420.

On the other hand, an upper bound of the critical radius may be determined by examiningthe function v (r , t) in a similar way. Since v (r , t) is a lower solution, if it quenches then so willu (r , t). Consider solutions to v (r , t) of the form

v (r , t) = g (t) J0 (γ r) ,

where J0 (r) is the zeroth-order Bessel function of the first kind and γ is the first nonzero root ofJ0 (r). Following the same derivation as in [14], it is observed that if � > γ/2 then the solutionu (r , t) of (2.4)–(2.7) will quench. Putting both lower and upper bound estimates together, wemay conclude that for the critical quenching radius of �,

�∗ ∈ (1.1421, 1.20241) . (2.8)

III. FULLY ADAPTIVE METHOD

Let r0 = 0, rN+1 = 1 for a sufficiently large N. Denote vj (t) = v(rj , t

)as an approximation of

the solution u at(rj , t

), in which rj = rj−1 +hj−1, j = 1, . . . , N +1, and hj−1 is the variable step

size between adjacent spatial nodes. Further, define rj+1/2 = rj + hj/2 and rj−1/2 = rj − hj−1/2.Utilizing central differences over the nonuniform spatial grids, we acquire from (2.4), (2.5) thesemidiscretized system,

v′j = 2α

rj

(hj + hj−1

)(

rj−1/2

hj−1vj−1 + rj+1/2

hj

vj+1 − hj rj−1/2 + hj−1rj+1/2

hjhj−1vj

)

+ f(vj

), j = 1, . . . , N , (3.1)

v′0 = 4

h20

α (v1 − v0) + f (v0) . (3.2)



A. Nonuniform Finite Difference Scheme

Equations (3.1), (3.2) along with (2.6), (2.7) can be compressed into matrix form, namely,

v′ = Pv + g (v) , (3.3)

v (0) = vinit, (3.4)

where v (t) = (v0, v1, . . . , vN)T , g (v) = (f (v0) , f (v1) , . . . , f (vN))T and vinit =(u0 (0) , u0 (r1) , . . . , u0 (rN))T . The matrix P = αT ∈ R

(N+1)×(N+1), where T is tridiagonal withlower, main, and upper diagonals of

lj = 2rj−1/2

hj−1

(hj−1 + hj

)rj

, j = 1, . . . , N ,

m0 = − 4

h20

, mj = − 2

hjhj−1, j = 1, . . . , N ,

u0 = 4

h20

, uj = 2rj+1/2

hj

(hj + hj−1

)rj

, j = 1, . . . , N − 1.

There are four advantageous properties of the matrix P. It is weakly diagonally dominant,similar to a symmetric matrix, the eigenvalues are negative, and is diagonalizable. Their proofsare straightforward and highlighted in the following lemmas.

Lemma 3.1. The matrix P is weakly diagonally dominant.

Proof. Let βj = ∑N+1i=1 Pj ,i . Clearly β1 = 0. For βj , j = 2, . . . , N ,

βj = lj + mj + uj = − 2

hjhj−1+ 2rj − hj−1

rjhj−1

(hj + hj−1

) + 2rj + hj

rjhj

(hj + hj−1

)

= − 2

hjhj−1+ 2hj−1 + 2hj

hjhj−1

(hj + hj−1

) = 0.

Finally, we consider βN +1.

βN+1 = lN + mN = 2

hN−1

⎛⎜⎝

rN − 1

2hN−1

(hN + hN−1) rN

− 1

hN

⎞⎟⎠

<2

hN−1

⎛⎜⎝

rN − 1

2hN−1

hNrN

− 1

hN

⎞⎟⎠ <

2

hN−1

(1

hN

− 1

hN

)= 0.

Therefore, βj ≤ 0 for all j.

Lemma 3.2. The matrix P is similar to a symmetric, tridiagonal matrix.

Proof. See similar discussions in [15].



FIG. 1. The directed graph of P, that is, the graph of N + 1 nodes Q1, Q2,...,QN +1 such that there is adirected arc or path from Qi to Qj if and only if Pi,j �= 0 [15, 16]. [Color figure can be viewed in the onlineissue, which is available at wileyonlinelibrary.com.]

Lemma 3.3. The eigenvalues of P are real and less than zero.

Proof. According to Lemma 3.2, all eigenvalues of P must be real. Recall the proof ofLemma 3.1. An application of Geršgorin’s circle theorem indicates that all eigenvalues of P mustbe nonpositive.

To show that zero cannot be an eigenvalue, amounts to showing that P is invertible. Recall thatthe main, upper, and lower diagonals of P are all nonzero. Consider a directed graph of P. Thedetails of the graph are shown in Figure 1.

Clearly, the graph is strongly connected, that is, there exists a directed path of finite lengthbetween any two distinct nodes. Therefore, as the matrix P is weakly diagonally dominant andhas a strongly connected directed graph, it is guaranteed to be invertible (see [16] §6.2.9).

Lemma 3.4. The matrix P is diagonalizable.

Proof. By Lemma 3.2, P is similar to a symmetric matrix, that is, P = YMY −1, where M isa symmetric matrix. Therefore, P and M share the same eigenvalues and, by the previous lemma,M is invertible. Therefore, M is also diagonalizable, and there exists a nonsingular matrix V anda diagonal matrix D such that

P = YV DV −1Y −1 = ZDZ−1.

Consequently, P is similar to a diagonal matrix and, hence, diagonalizable.

The semidiscretized system (3.3)–(3.4) can be solved exactly, that is,

v (t) = eP tv0 +∫ t

0e(t−τ)P g (v) dτ , t ≥ 0.

From a numerical perspective, this equation can be approximated via a suitable quadrature. Inthis study, a trapezoidal rule is considered. This leads to a second order formula,

v (t) = eP t

(v0 + t

2g (v0)

)+ t

2g (v (t)) + O

(t2

), 0 ≤ t � 1.

Further, a [1/1] Padé approximation of the matrix exponential yields

v (t) =(

I − t

2P

)−1 (I + t

2P

) (v0 + t

2g (v0)

)+ t

2g (v (t)) + O

(t2

), 0 ≤ t � 1.



The above suggests a second-order fully discretized scheme

vk+1 =(I − τk

2P

)−1 (I + τk

2P

) (vk + τk

2g (vk)

)+ τk

2g (wk) , (3.5)

where τk = tk+1 − tk , k = 0, 1, 2, . . . , is a variable temporal step size and wk is a suitable approx-imation of vk+1. For the sake of higher computability in applications, choices of wk include anyfirst-order method such vk+τk (P vk + g (vk)) via the forward Euler method. This particular choiceavoids solving a nonlinear system at each time step while maintaining the order of accuracy intime [6, 13].

As has been pointed out in recent publications, temporal and spatial adaptations plays a criticalrole in capturing the solution’s most delicate behavior near the onset of quenching [3, 6, 10].Moving-mesh strategies are frequently used to determine variable temporal grids consistent tosolution profiles; however, it has been observed that this methodology may complicate the cal-culations as evolution equations are coupled with that of the mesh equations [17]. The approachof this article capitalizes on independent adaptations through equidistribution principles throughdifferent monitoring functions in time and space, respectively. Once temporal and spatial gridsare determined at a certain level, they are used for the next step in the calculation of the solutionv in (3.5).

B. Temporal Adaptation

The temporal adaptation utilized is implemented via an arc-length monitor of the solution functionv(t) along the characteristic lines. It is significantly different from most existing approaches [5].In this process, a temporal step is chosen such that the arc-lengths between successive iterates areequal. More precisely, we require that

((v′

k+1 − v′k

) · ej

)2 +(τ

(j)

k

)2 = ((v′

k − v′k−1

) · ej

)2 +(τ

(j)

k−1

)2, 1 ≤ j ≤ N + 1,

where vk is the solution vector of (3.5) at the kth temporal level, ej ∈ RN+1 is the jth unit vector,

and τ0 > 0 is given. Because the above implicit equation can be solved along each of the N + 1characteristic lines, the temporal step is chosen as the minimum value, that is,

τk = min0≤j≤N

τ(j)

k , k = 0, 1, 2, . . .

It is clear that the above procedure requires information from the numerical solution at multipletemporal levels. This is not a challenge as the first few temporal advancements of the numericalsolution rarely require any adaptations. Several sufficiently small uniform initial temporal stepsare often adequate until the necessary information becomes available.

C. Spatial Adaptation

Spatial adaption has presented unique challenges in quenching computations. In particular, therehave been disagreements between selection criteria for ideal monitoring, or target, functions thatare to be equidistributed. Traditional selections tend to be the computed numerical solution or aparticular derivative. This often leads to overrefinement, that is, to a large disparity between thelargest and smallest grid sizes. As a result, this may impair the accuracy and lead to nearly sin-gular matrices or unexpected interpolation errors [13]. Hence, an improved approach has become



necessary for solving quenching-type reaction-diffusion equations over circular domains [11].Inspired by the recent development of EEG adaptive schemes, a target function is developed viaan exponential function that approximates the physical solution in a least squares sense. This for-mulation takes advantage of knowledge of known quenching profiles and generates adaptive gridsthat are consistent with the developing quenching solution. Moreover, the grids are developed ina smooth manner and the disparity of grid sizes can be controlled.

Let q(r) be a modified target function. Its arc-length over the polar radius can be calculated via

L =∫ 1

0

√1 + q2

r dr .

Consider the exponential target function q (r , t) = κ (t) eγ (t)r to be fitted to the physical solu-tion u (r , t) in the least squares sense. Hence, the equidistribution formula determining new gridlocations [6, 9] at time t becomes

1

N

∫ 1

0

√1 + (γ κeγ r)2dr =

∫ ri+1

ri

√1 + (γ κeγ r)2dr , i = 1, . . . , N .

The modified approach is advantageous in three essential ways. First, it emphasizes the alloca-tion of grid points near the quenching location. Second, the arc-length remains bounded and theadaptation can continue till quenching occurs. Third, the equidistribution equations use asymp-totic knowledge of the quenching profile of a numerical solution rather than the solution or itsderivatives directly. In all, this methodology smoothly adapts the grid points such that the dis-parity between grid sizes is controlled. This allows adaptation to continue throughout the entirecomputation, in particular up to the moment of quenching. The completion of grid adaptation ismade through the utilization of cubic spline interpolation of the current solution data on to thenewly formed grid. The involved matrices are then updated.

IV. MONOTONICITY, POSITIVITY, AND STABILITY

Positive solutions of quenching-type nonlinear problems (2.1)–(2.3) and (2.4)–(2.7) are knownto increase monotonically till a steady state or quenching is reached. Hence, it is of the upmostimportance for the numerical solution to capture these essential features. This hinges on establish-ing requirements on the temporal and spatial step sizes so that the coefficient matrices involvedremain positive [5, 18]. The positivity of the matrices is then used to verify the monotonicity ofthe numerical solution. The process often introduces additional constraints to adaptive temporaland spatial grid sizes.

Lemma 4.1. If

τk <1

αmin

{1

2h2

0, mini

{hihi−1}}

, i = 2, . . . , N + 1,

then I + τk2 P is positive and I − τk

2 P is inverse positive.

Proof. Recall Lemmas 3.1–3.4. The matrix I − τk2 P must be strictly diagonally dominant

and invertible. Note that(I − τk

2 P)i,i

> 0 and∑N+1

j=1

(I − τk

2 P)i,j

> 0 for all 1 ≤ i ≤ N + 1.

Hence, I − τk2 P satisfies the weak row sum criterion. Therefore, it is inverse positive [18].



Now, let us consider the elements of I + τk2 P . The off-diagonals are clearly positive according

to the definition of P. Consider the main diagonal entries,(I + τk

2P

)1,1

= 1 − 2τk

α

h20

> 0,

(I + τk

2P

)i,i

= 1 − τk

α

hi−1hi−2> 0, i = 2, . . . , N + 1,

provided that τk satisfies the indicated hypothesis. This completes our proof.

Lemma 4.2. Let vinit ≡ 0. If

τ0 <1

αmin

{1

2h2

0, mini

{hihi−1}}

, i = 2, . . . , N + 1,

then v1 > 0 and Pv0 + g0 > 0 , where the positivity of vectors is defined componentwise.

Proof. By means of (3.5), we have

v1 = τ0

2

(I − τ0

2P

)−1 (I + τ0

2P

)g0 + τ0

2g0 = τ0

(I − τ0

2P

)−1g0 > 0.

Lemma 4.3. Let vinit ≡ 0 and τ = τk , k = 1, 2, . . . If

τ <1

αmin

{1

2h2

0, mini

{hihi−1}}

, i = 2, . . . , N + 1,

and Pvk0 + gk0 > 0 for certain k0 ≥ 0, then vk+1 − vk > 0 for all k ≥ k0.

Proof. Observe that Lemma 4.2 guarantees the validity of the hypothesis Pvk0 + gk0 > 0 atleast initially. It is thus clear that vk0+1 − vk0 > 0 for such a k0 ≥ 0.

Assume that vk − vk−1 > 0, k ≥ k0. Thus,

vk+1 − vk =(I − τ

2P

)−1 (I + τ

2P

) (vk + τ

2g (vk)

)+ τ

2g (wk)

−[(

I − τ

2P

)−1 (I + τ

2P

) (vk−1 + τ

2g (vk−1)

)+ τ

2g (wk−1)

]

=(I − τ

2P

)−1 (I + τ

2P

) (vk − vk−1 + τ

2(gk − gk−1)

)

+ τ

2(g (wk) − g (wk−1)) > 0, k ≥ k0, (4.1)

due to the matrix positivity ensured by Lemma 4.1. Hence, the proof is completed through standardmathematical induction.

Lemma 4.4. Let v0 ≡ 0 and τk be variable. If

τk <1

αmin

{1

2h2

0, mini

{hihi−1}}

, i = 2, . . . , N + 1,



and Pvk0 +gk0 > 0 for certain k0 ≥ 0, then vk+1 −vk > 0 for all k ≥ k0 when τk is sufficientlysmall.

Proof. Similar to that in Lemma 4.3, we readily have vk0+1 − vk0 > 0. Assume thatvk − vk−1 > 0, k ≥ k0. It follows from (3.5) that

vk+1 − vk =(I − τ

2P

)−1 (I + τ

2P

)(P vk + gk) + O

(τ 2

k

), k ≥ k0. (4.2)

In fact, the vector vk+1 − vk is positive for sufficiently small τk . To see this,

Pvk + gk = P (vk − vk−1) + Pvk−1 + gk − gk−1 + gk−1

= gk − gk−1 + (P vk−1 + gk−1)

+ τk−1P(I − τk−1

2P

)−1(P vk−1 + gk−1) + O

(τ 2

k−1

)

= gk − gk−1 +[I + τk−1P

(I − τk−1

2P

)−1]

(P vk−1 + gk−1) + O(τ 2

k−1

)

= gk − gk−1 +[(

I − τk−1

2P

)+ τk−1P

]

×(I − τk−1

2P

)−1(P vk−1 + gk−1) + O

(τ 2

k−1

)

= gk − gk−1 +(I + τk−1

2P

) (I − τk−1

2P

)−1

× (P vk−1 + gk−1) + O(τ 2

k−1

)> 0,

provided that τk−1 is sufficiently small. Hence Pvk + gk > 0 and vk+1 − vk > 0 provided that τk

is sufficiently small.

The lemmas lead immediately to the following theorem that provides the necessary criteria toensure the positivity and monotonicity of the numerical solution.

Theorem 4.1. Let � ≥ 0 be any beginning temporal level considered. Assume that

τk <1

αmin

{1

2h2

0, mini

{hihi−1}}

, k ≥ �,

for all available indexes i. If Pv� +g� > 0 and τk , k ≥ � is sufficiently small, then the sequence{vk}k≥� generated by the fully adaptive method (3.5) remains positive and increases monotonicallyuntil unity is exceeded by a component of the solution vector vk or converges to the steady solutionof the problem (2.4)–(2.7) for both variable and constant temporal steps τk , k ≥ �.

Numerical stability has been a sensitive issue for quenching-type nonlinear problems[5, 9, 10, 13]. However, it has been observed that if the solution varies relatively slowly, an insta-bility can be detected through a conventional linear stability analysis of the nonlinear differenceschemes involved [13, 19]. Moreover, the temporal adaptation discussed in Section 3.2 provides



an avenue to control the growth in the solution. Although applications of such analysis to non-linear equations cannot be rigorously justified, it is found to be easy to use in practice and mayensure certain boundedness of the perturbed solution [20].

To this end, we shall focus on the stability of the fully adaptive scheme (3.5) with a frozennonlinear term. Freezing the source term g is equivalent to assuming that it does not depend onthe solution. Because the linear part of g may dominate the Laplacian term near quenching, thisis a limitation of the current analysis. But nevertheless, numerical experiments indicate that themonotonicity-preserving numerical scheme on nonuniform meshes is indeed stable.

Theorem 4.2. The fully adaptive method (3.5) with its nonlinear source term frozen is implicitand stable in the von Neumann sense.

Proof. Assume that a finite number of spatial and temporal adaptations will be performedthroughout the duration of the computation. Let β be the finite number of temporal and spatialgrid pairs used. Each pairing is used at least once in the computation. Let Mi be the number ofiterates that each pairing is used in the computation for i = 1, . . . , M . Without loss of generality,let Si be the amplification matrix after Mi multiplications, that is,

Si =[(

I − τi

2Pi

)−1 (I + τi

2Pi

)]Mi

.

Recall from Lemma 3.4 that for each i the matrix Pi is diagonalizable, hence,

Si = Zi

[(I − τi

2Di

)−1 (I + τi

2Di

)]Mi

Z−1i = Zi(r11 (τiDi))

Mi Z−1i ,

where Di is a diagonal matrix with negative entries, guaranteed by Lemma 3.3, and r11 (τiDi) isthe [1/1] Padé approximant. Therefore, the spectral norm of Si is uniformly bounded by a constant,ci . Now,

∥∥∥∥∥M∏i=1

Si

∥∥∥∥∥2

≤M∏i=1

ci ≤ c,

where c = max1≤i≤Mci . Therefore, ||zk+1||2 ≤ c||zinit||2, where zinit = vinit − vinit is an initialperturbation and zk+1 = vk+1 − vk+1 is the perturbation arising from the initial perturbation [15].Therefore, the theorem is clear.

V. NUMERICAL EXPERIMENTS

Consider the quenching-type reaction-diffusion problem (2.1)–(2.3). Because numerical explo-rations in the field are still open and new, our main focus will be on three interconnected andfundamental issues. First, we will validate and refine numerically the theoretical prediction (2.8).An illustrative depiction of the exponential evolving grids will be given over the circular domain.Then, in the second experiment, characteristics of the evolving grids will be presented andillustrated. Experimental quantifications of the mesh movement will be provided for the one-dimensional (1D) mesh. As suggested in earlier work [11], the experiments again show thatidentification of the quenching location occurs early while refinement near that location contin-ues through final stages of the computation. Though this may not indicate a rigorous justification



FIG. 2. LEFT: Outcomes of the quenching time with respect to different � values. It is interesting to observethat T∗ increases beyond exponentially fast as � → 1.1421+ and T∗ → 0.5 as � increases. Logarithmicscales are used. RIGHT: A plot of the last EEG mesh used immediately before quenching for the case of� = 1.14500. N = 200 is used (only about 1/3 of the grid-points are shown for a better visual clarity). [Colorfigure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

of the underlying adaptation methodology, it does provide experimental evidence and confidencefor this and further exploration. Finally, in the third experiment, the impact of a nonconventionalnonlinear memory source function [12] will be investigated.

A. Example 1

Let f (u) = 1/ (1 − u) , 0 ≤ u < 1. In this case, we recall (2.8). The fully adaptive finite dif-ference scheme (3.5) is used to experimentally verify if steady state or quenching solutions areobserved for radii � less than 1.1421 and greater than 1.20241, respectively. The computationsare carried out with a relatively fine spatial mesh containing N = 200 grid points. The initialgrid is uniform and adaptations are performed through the EEG principal. Temporal adaptationprocedures are also incorporated.

The critical quenching radius is refined by creating a sequence of experiments with decreas-ing radii beginning from a known quenching radius close to 1.14210. The final critical radiusis determined to be the smallest possible radius for which numerical quenching remains. In thisway, a least critical radius of �∗ = 1.14450 is observed, notably just slightly above the theoreticalestimation � = 1.14210. Therefore, there is a high degree of confidence in the sharpness of (2.8)’slower � bound for quenching. The quenching times for � ∈ [1.1445, 3.0] together with the finalspatial mesh used in the case of � = 1.14450 are shown in Figure 2. Experimental results forselected � values from the interval [1.14241, 1.203] are given in Table I. Interestingly, for largerradii, the quenching time tends to 0.5. This corresponds to the spatially homogenous solution’squenching time over the infinite domain with a zero initial condition.

B. Example 2

Again, let f (u) = 1/ (1 − u) , 0 ≤ u < 1, and set the initial uniform spatial step h = 0.01and temporal step τ0 = 10−5. Take � = 2, therefore a quench should occur according to(2.8) and Example 1. Computations continue on the uniform grids till maxru (t) = 0.1 wherethe spatial adaptation mechanism kicks in automatically. In this experiment, subsequent spatial



TABLE I. Experimental results for selected � values covering the interval given in (2.8).

� maxr v(T∗) maxr v t (T∗) T∗

1.14241 0.60907 <10−6 NA1.14250 0.60992 <10−6 NA1.14450 0.99945 428.80412 43.7652201.14500 0.99914 134.85705 17.9467551.15000 0.99955 442.86227 5.6539401.15500 0.99773 241.85236 4.0313701.16000 0.99912 386.56015 3.2794301.16500 0.99948 435.06027 2.8253201.20000 0.99919 407.79288 1.6611401.20241 0.99907 384.72506 1.6242501.20300 0.99982 489.21327 1.615590The absolute maximum values of the numerical solution and temporal derivative at the completion of the numericalcomputation are also listed, in addition to the quenching times T∗, where applicable.

adaptations are conducted whenever maxru (t) = 0.1 + 0.005j < 1. This renders a quenchingtime T ∗ ≈ 0.53651948547694. Cubic splines are utilized in interpolating the current solutiondata to the newly formed grids.

Figure 3 depicts the dynamic mesh throughout the computation. A collection of actual spatialstep sizes selected at certain temporal levels are given in the first picture. Sample data are takenfrom temporal levels corresponding to max u = 0.1k, k = 0, 1, 2, . . . , 9. Although the initialuniform steps are shown as a constant h = 0.01 the distribution begins to twist as soon as theadaptation kicks in. In the end, the minimal spatial step size is approximately h0 = 0.0053. Notethat the distribution of the spatial step sizes is monotone as the minimum of hi is always locatednear the quenching location at the origin. The need for mesh adaptation, or refinements, is greatlyincreased in the later stage of quenching computations, as predicted in [11]. The second frameshows a profile of the minimum spatial size, h0, throughout the computation. Note the change ofh0 is rather smooth and tame in the beginning, while it becomes faster throughout the adaptation

FIG. 3. LEFT: The evolution profile of the spatial discretization parameters {hi}200i=1 is shown. Curves

are shown for max u = 0.1k, k = 0, 1, 2, . . . , 9, respectively indicated from high (violet) to low (red)frequency. RIGHT: A plot of the first spatial step h0 versus time t. This is the location of the finest resolu-tion nearest the quenching location. [Color figure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]



FIG. 4. LEFT: A simulation of the numerical solution v immediately before a quench (maxr ,t v ≈0.99515302842726) at t = 0.53651948547694. RIGHT: The corresponding derivative function vt priorto quenching (maxr ,t vt ≈ 956.4740096070291). [Color figure can be viewed in the online issue, which isavailable at wileyonlinelibrary.com.]

till the onset of quenching where the fastest reductions are observed. Again, while this does notprovide a rigorous justification, it does present evidence that the grid mapping function is smoothin this case, yielding higher accuracy in the location truncation error [17]. Finally, 3D figures ofthe numerical solution v and its temporal derivative vt immediately before quenching are givenin Figure 4.

C. Example 3

Forcing terms of Eqs. (2.1)–(2.3) or (2.4)–(2.7) may involve nonlocal reaction terms [21]. Nonlo-cal reaction terms model the induction period of the thermal explosion process for a compressiblereactive gas [22, 23]. The source term may involve an integration of the temperature and itsderivatives over the spatial domain. On the other hand, the integration may be over the temporalcoordinate as found in quenching models of population dynamics [12, 24, 25]. Here, an exampleof a nonlocal source term is

f (u) = κ

(1 − u)p

∫ t

0

dτ

(1 − u)q , 0 ≤ u < 1, (5.1)

where q, κ > 0 and p ≥ 0. For illustration purposes, we consider a particular application withκ = p = 1, q = 3, and � = 2 [1, 12, 14]. The fully adaptive scheme (3.5) is used to approximatethe true solution with (5.1). A uniform initial spatial mesh with a uniform step size h = 0.001 isused. In such case, a numerical quenching time T ∗ = 0.8952 is acquired.

A distribution of the underlying adaptive mesh steps in space, together with the smoothnessratio ρi = hi+1/hi , i = 0, 1, 2, . . . , N , are shown in Figure 5. Again, the distribution is monotonewith the finest step near the quenching location, that is, the origin. The ratio of 1 < ρi < 1.015ensures the monotonic increasing of the spatial grids as r increases and indicates the grid issmoothly adapted [9, 10]. Three-dimensional plots of the numerical solution v and its temporalderivative vt at t∗ ≈ 0.86925 immediately before quenching are given by Figure 6. The simulationsdemonstrate precisely the expected quenching-combustion features. Finally, we show 2D snap-shots of the last four numerical solutions in the radial coordinate at temporal levels t = 0.869099,0.869149, 0.86919, 0.86925, just prior to quenching, as well as their corresponding temporal



FIG. 5. LEFT: Snapshots of the adaptive spatial step sizes at k = 1,468 (black, blue, respectively) andimmediately before the quench (red). RIGHT: Distributions of the corresponding smoothness ratios of theadaptive step sizes prior to the quench. [Color figure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]

derivatives in Figure 7. A quenching blow-up is clearly noticeable in the 2D shots. Interestingly,the profiles of v indicate a steeper gradient as compared with typical solution profiles obtainingwith the standard forcing term f (u) = 1/ (1 − u) . Although the nonlinear memory in (5.1) helpsto increase the quenching time as compared with that in Example 2, it does promote a faster rateof change in the solution’s gradient near the quenching location.

Interestingly, in this particular case, max u = 0.2 at t = 0.58249. Since quenching occurs att = 0.86925 then it takes approximately 67% of the calculation to achieve a 0.2 increase in themaximum value of v as compared to its initial value. In contrast, max u = 0.9 at t = 0.868249,from that point it takes 0.01% of remaining calculation to obtain quenching and achieve its max-imum value. The tame growth of the solution stems from an initially small initial condition and,consequently, the integral in (5.1) is quite small. This depresses the forcing term early in thecomputation. However, once the solution grows to a sufficient size, this is no longer the case andthe value of the integral becomes remarkably large. Hence, the quenching singularity becomesstronger in the end.

FIG. 6. LEFT: A simulation of the numerical solution v prior to a quench at t = 0.86924999999991(maxr ,t v ≈ 0.98603524399047). RIGHT: The corresponding derivative function vt prior to quenching(maxr ,t vt ≈ 387.6066161997471). [Color figure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]



FIG. 7. LEFT: Two-dimensional snapshots of the numerical solution v in the radial coordinate at temporallevels t = 0.869099, 0.869149, 0.86919, 0.86925, just before quenching. RIGHT: Profiles of the temporalderivative vt at t = 0.869149, 0.86919, 0.86925. A rapid increase in the solution and its derivative near theonset of the quenching are observable. The red curves show the solution and temporal derivative just priorto quenching. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

VI. CONCLUSIONS

This article concerns a fully adaptive finite difference scheme for solving quenching-type reaction-diffusion equations over circular domains. The equations have important applications in modelingmultiphysics processes, in particular combustion. The initial conditions are assumed to be radi-ally symmetric so that the θ -dependence can be neglected. An auxiliary condition at the originis introduced to enable the computation and analysis of the constructed algorithm. The monot-onicity, positivity, and stability of the numerical solution are ensured. Theoretical bounds of thecritical quenching domain radius in a standard case are derived and validated via computationalexperiments. The fully adaptive scheme implemented is also tested on an exploratory problemwith a nonconventional nonlinear memory source term. It is observed experimentally that whilethe quenching time is increased the quenching event is more explosive in problems with nonlinearmemories as compared with those without them.

The temporal adaption follows a conventional type arc-length monitoring approach. However,the spatial adaption uses a more sophisticated EEG monitoring system, in which an exponentialfunction is fitted to the target function profile in the least squares sense. Consequently, a bal-anced temporal-spatial adaptation mechanism is accomplished for the computation. This novelmethodology has four distinct advantages as compared to any conventional procedure. First, themesh generated will be the finest possible near areas of the greatest solution growth, namely thequenching location. The discretization step sizes increase smoothly as one moves away from sen-sitive locations. Second, arc-lengths of new monitoring functions do not tend to infinity allowingfor adaptation to occur throughout the entire computational domain. Third, a priori informa-tion of quenching solution profiles is used to conveniently develop the modified target function.Fourthly, the EEG procedure constructed via least squares approximations provides a desirablemesh smoothness throughout the adaptive computation. Therefore, the chances of nonphysicaloscillations are significantly reduced. The numerical experiments provide evidence that theseadvantages are realized in the computation. Indeed, the EEG procedure is new and its realizationsfor different equations and under different environmental settings still need to be implemented



and studied, it has provided a substantial impact to the current theory and practice of adaptivemethods. Interested readers are referred to more detailed discussions in [6] and references therein.

A nonlinear stability analysis is still to be fulfilled. Implementations of such an analysis viaweighted conservations of the singular solution are currently being investigated. The initial con-cept used has been discussed in [2, 5, 20]. Further, our explorations have been extended to thealgebraic stability in connection with applications of the latest theory of geometric integrators[5, 11, 13].

There are two immediate extensions of this article, for which the authors have been diligentlyworking toward. First, more general initial conditions and inclusions of mathematical degenera-cies [10] will be used. Although in the circumstance the radially symmetry of the solution canno longer be assumed, variational EEG structure designs have been in a good progress. Second,the analysis acquired for circular domains can be carried over to cases with similar geometries,in particular elliptical domains, for more realistic applications. Both extensions are of the upmostimportance for future explorations.

The authors would like to thank the anonymous referees for their valuable comments and sug-gestions that have enhanced the quality and presentation of this paper. The second author alsoappreciates the supports and encouragements from the U.S. Air Force Research Laboratory andBaylor URC Council.

References

1. J. Bebernes and D. Eberly, Mathematical problems from combustion theory, Springer-Verlag, Berlin,New York, 1989.

2. T. Poinsot and D. Veynante, Theoretical and numerical combustion, Edwards Publisher, Philadelphia,2005.

3. M. A. Beauregard and Q. Sheng, An adaptive splitting approach for the quenching solution ofreaction-diffusion equations over nonuniform grids, J Comput Appl Math 241 (2013), 30–44.

4. C. Y. Chan, A quenching criterion for a multidimensional parabolic problem due to a concentratednonlinear source, J Comput Appl Math 235 (2011), 3724–3727.

5. Q. Sheng, Adaptive decomposition finite difference methods for solving singular problems, FrontiersMath China 4 (2009), 599–626.

6. M. A. Beauregard and Q. Sheng, Solving degenerate quenching-combustion equations by an adaptivesplitting method on evolving grids, Comput Struct 122 (2013), 33–43.

7. C. Kirk and W. Olmstead, Blow-up in a reactive-diffusive medium with a moving heat source, Z AngewMath Phys 53 (2002), 147–159.

8. W. Wang and S. Zheng, Asymptotic estimates to quenching solutions of heat equations with weightedabsorptions, Asymptotic Anal 70 (2010), 125–139.

9. Q. Sheng and A. Q. M. Khaliq, A compound adaptive approach to degenerate nonlinear quenchingproblems, Numer Methods Partial Differ Equ 15 (1999), 29–47.

10. H. Cheng, P. Lin, Q. Sheng, and R. C. E. Tan, Solving degenerate reaction-diffusion equations viavariable step Peaceman-Rachford splitting, SIAM J Sci Comput 25 (2003), 1273–1292.

11. M. A. Beauregard and Q. Sheng, Explorations and expectations of equidistribution adaptations fornonlinear quenching problems, Adv Appl Math Mech 5 (2013), 407–422.

12. S. Zhou, C. Mu, Q. Du, and R. Zeng, Quenching for a reaction-diffusion with nonlinear memory,Commun Nonlinear Sci Numer Simul 17 (2012), 754–763.

13. Q. Sheng and A. Q. M. Khaliq, A revisit of the semi-adaptive method for singular degeneratereaction-diffusion equations, East Asia J Appl Math 2 (2012), 185–203.



14. W. Walter, Parabolic partial differential equations with a singular nonlinear term, Funkcial Ekvac 19(1976), 271–277.

15. G. Golub and J. Ortega, Scientific computing and differential equations: an introduction to numericalmethods, Academic Press, Boston, 1992.

16. R. Horn and C. Johnson, Topics in matrix analysis, Cambridge University Press, London, New York,1991.

17. C. Budd, W. Huang, and R. D. Russell, Adaptivity with moving grids, Acta Numer 18 (2009), 111–241.

18. P. Henrici, Discrete variable methods in ordinary differential equations, Wiley, New York, 1962.

19. E. H. Twizell, Y. Wang, and W. G. Price, Chaos-free numerical solutions of reaction-diffusion equations,Proc R Soc Lond Sect A 430 (1991), 541–576.

20. M. Schatzman, Stability of the Peaceman-Rachford approximation, J Funct Anal 162 (1999), 219–255.

21. P. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J Math Anal 29 (1998), 1301–1334.

22. J. Bebernes and A. Bressan, Thermal behavior for a confined reactive gas, J Differ Equ 44 (1982),118–133.

23. W.E. Kastenberg, Space dependent reactor kinetics with positive feed-back, Nukleonika 11 (1968),126–130.

24. C. A. Roberts, Recent results on blow-up and quenching for nonlinear Volterra equations, J ComputAppl Math 205 (2007), 736–743.

25. P. Souplet, Monotonicity of solutions and blow-up for semilinear parabolic equations with nonlinearmemory, Z Angew Math Phys 55 (2004), 28–31.


exponential time differencing crank–nicolson method with a quartic spline approximation for...

Documents