Higher-order diagonal Padé nonlinear absorbing boundary conditions for nonlinear time-fractional parabolic problems on unbounded domains

For the nonlinear time-fractional parabolic Cauchy problem on the real line, the only known rigorously analyzed absorbing boundary conditions (ABCs) that explicitly incorporate the nonlinear reaction term are built from the first diagonal Padé approximation of the Laplace-domain symbol $\sqrt{s^{\alpha}}$. The 2016 source paper that introduced this construction explicitly posed the derivation of higher-accuracy nonlinear ABCs from higher-order diagonal Padé expansions as an open problem: for approximation orders $N>1$, clearing denominators brings fractional time derivatives of composite order $2\alpha$ and higher, acting on both the solution and the nonlinear term, into the boundary operator, and the algebraic reduction that closes the $N=1$ case into auxiliary fractional evolution equations no longer applies. No higher-order nonlinear Padé absorbing boundary condition has been derived or analyzed in the literature to date, so the problem remains open.

Background

Equations of the form

$${}_0^C D_t^\alpha u(x,t)=u_{xx}(x,t)+f(u(x,t)),\quad x\in\mathbb{R},\ 0<t\le T,\ 0<\alpha<1,$$

with compactly supported initial data $u(x,0)=u_0(x)$ and $u\to 0$ as $|x|\to\infty$, model anomalous subdiffusion — diffusive transport in which the mean-square displacement grows more slowly than linearly in time — with memory, e.g. Fisher-type and Huxley-type reaction dynamics, two classic nonlinear reaction-diffusion models with logistic and bistable reaction kinetics respectively. Here ${}_0^C D_t^\alpha$ is the Caputo fractional derivative, defined by ${}_0^C D_t^\alpha u(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t(t-\sigma)^{-\alpha}\frac{du}{d\sigma}\,d\sigma$, whose nonlocality in time encodes the memory effect. Because the spatial domain is unbounded, a direct simulation must truncate it to a finite interval $[l_-,l_+]$; imposing naive boundary conditions there reflects outgoing disturbances back into the computational domain and pollutes the solution. Absorbing boundary conditions (ABCs, also called artificial boundary conditions) are boundary relations imposed at the artificial boundaries $x=l_\pm$ that let outgoing disturbances leave with minimal spurious reflection.

For the linear part ${}_0^C D_t^\alpha u=u_{xx}$, a Laplace transform in time (with transform variable $s$) yields exterior solutions behaving like $e^{\mp\sqrt{s^{\alpha}}|x|}$, and the exact nonreflecting condition involves the nonlocal (convolutional) operator with symbol $\sqrt{s^{\alpha}}$. A standard way to make such a condition local and implementable is to approximate $\sqrt{s^{\alpha}}$ by a diagonal Padé approximation — a rational function of $s^{\alpha}$ whose numerator and denominator have equal degree $N$ — centered at a chosen expansion point $s_0>0$ (interpretable as a dominant frequency squared). Li and Zhang (2016) took the first diagonal Padé approximation

$$\sqrt{s^{\alpha}}\approx s_0^{\frac{\alpha}{2}}\,\frac{s_0^{\alpha}+3s^{\alpha}}{3s_0^{\alpha}+s^{\alpha}},$$

substituted the resulting one-directional (one-way) operator into the full nonlinear equation, and used the equation itself to eliminate the fractional derivative, obtaining the nonlinear ABC

$$\Bigl(\partial_x\pm 3s_0^{\frac{\alpha}{2}}\Bigr){}_0^C D_t^\alpha u+\Bigl(3s_0^{\alpha}\partial_x\pm s_0^{\frac{3\alpha}{2}}\Bigr)u=\Bigl(\partial_x\pm 3s_0^{\frac{\alpha}{2}}\Bigr)f(u)\quad\text{at }x=l_\pm.$$

Introducing auxiliary boundary variables this closes into a system of Caputo evolution equations on the bounded interval, which the authors discretized by a linearized finite-difference scheme (an L1-type scheme, i.e. a piecewise-linear-in-time approximation of the Caputo derivative, with the nonlinearity evaluated at the previous time level so that no per-step nonlinear iteration is needed), proving unconditional boundedness and the global error bound $O(h^2+\tau)$, where $h$ and $\tau$ are the space and time steps. The paper also records the general-$N$ Padé error formula

$$E_N(s^{\alpha})=2\sqrt{s^{\alpha}}\,\frac{\gamma^{2N+1}}{1+\gamma^{2N+1}},\qquad\gamma=\frac{\sqrt{s^{\alpha}}-s_0^{\frac{\alpha}{2}}}{\sqrt{s^{\alpha}}+s_0^{\frac{\alpha}{2}}},$$

showing that larger $N$ approximates $\sqrt{s^{\alpha}}$ more accurately away from $s_0$.

The unresolved difficulty is the passage to $N\ge 2$. After clearing the higher-degree denominators, the boundary operator contains higher powers of $s^{\alpha}$, i.e. Caputo derivatives of composite orders such as $2\alpha$ (which exceeds $1$ when $\alpha>\tfrac12$), acting on both $u$ and the nonlinear term $f(u)$; the elimination trick that closed the $N=1$ condition no longer applies, and it is not known how to reduce such operators to an implementable, analyzable system. The linear equation does admit high-order local ABCs of this type: Zhang, Li, Wu and Zhang (2017) used Padé expansion plus auxiliary variables that reduce the order of the time derivatives to build stable, convergent high-order local ABCs for the linear fractional diffusion equation — evidence that the obstruction is genuinely created by the coupling with $f(u)$. Subsequent nonlinear constructions (time-fractional Burgers, time-fractional Kardar–Parisi–Zhang, nonlinear fractional diffusion with delay) have all reused the $N=1$ operator, and fast sum-of-exponentials evaluation of the Caputo history — compression of the convolution kernel inside the Caputo derivative into a short sum of decaying exponentials, so that the per-step cost of evaluating the history term no longer grows with the number of elapsed time steps — as well as high-order time stepping are well developed for linear and bounded-domain problems, leaving the higher-order nonlinear ABC construction as the specific gap.

Problem Statement

Construct, and rigorously analyze, higher-accuracy nonlinear absorbing boundary conditions derived from higher-order diagonal Padé expansions of the Laplace-domain symbol $\sqrt{s^{\alpha}}$ for the nonlinear time-fractional parabolic Cauchy problem

$${}_0^C D_t^\alpha u=u_{xx}+f(u),\quad x\in\mathbb{R},\ 0<t\le T,\ 0<\alpha<1,$$

with compactly supported initial data and $u\to 0$ as $|x|\to\infty$, generalizing the known $N=1$ construction of Li and Zhang.

Specifically, for approximation orders $N\ge 2$ (the diagonal Padé family whose $N=1$ member is $\sqrt{s^{\alpha}}\approx s_0^{\alpha/2}(s_0^{\alpha}+3s^{\alpha})/(3s_0^{\alpha}+s^{\alpha})$ at expansion point $s_0>0$), clearing denominators brings fractional time derivatives of composite orders such as $2\alpha$ - acting on both the solution $u$ and the nonlinear term $f(u)$ - into the boundary operator, and the algebraic reduction that closes the $N=1$ case into auxiliary Caputo evolution equations no longer applies. The construction and analysis require:

(i) give the explicit boundary operators at the artificial boundaries $x=l_\pm$ and close them into a well-posed initial-boundary value problem on $(l_-,l_+)$ whose solution approximates the Cauchy solution, with a boundary symbol approximation of strictly higher order than the $N=1$ condition in the sense of the Padé error $E_N$ defined in the Background;

(ii) provide a practical time-stepping scheme for the coupled interior-boundary problem; and

(iii) prove stability and a global error bound for the coupled scheme in which the boundary-induced error is of strictly higher order in the Padé order $N$ than in the $N=1$ construction, measured against the $O(h^2+\tau)$ benchmark of the $N=1$ linearized scheme; the benchmark is a comparison reference for the boundary error rather than a rate the overall $h,\tau$ error must beat, unless the interior discretization is upgraded beyond the $N=1$ L1-type scheme. State how the boundary error term and the overall $h,\tau$ rate depend on $\alpha$, $N$, and $s_0$.

References

1. Dongfang Li, Jiwei Zhang, "Efficient implementation to numerically solve the nonlinear time fractional parabolic problems on unbounded spatial domain", Journal of Computational Physics 322 (2016) 415-428, doi:10.1016/j.jcp.2016.06.046, https://doi.org/10.1016/j.jcp.2016.06.046 (source paper)
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