Koleva and Vulkov (2007) derived exact artificial boundary conditions that reduce the heat equation with nonlinear boundary conditions on a semi-infinite line and a semi-infinite strip to bounded computational domains, but their implicit-explicit Euler discretization must store and reprocess the entire boundary history, because the boundary conditions contain temporal convolutions with weakly singular kernels (1/t−λ1/\sqrt{t-\lambda} in 1D; e−α(t−λ)/t−λe^{-\alpha(t-\lambda)}/\sqrt{t-\lambda} per strip mode in 2D). The authors state that they cannot find local-in-time recursive updates for these integrals and leave this, together with provably stable higher-order and adaptive time stepping, efficient long-time integration in the blow-up regime, and exact boundary conditions beyond these special geometries, open. Fast convolution methods developed since then (fast and oblivious convolution quadrature; sum-of-exponentials compression of power-law kernels) resolve the kernel-level recursion obstacle but have never been coupled to, or analyzed for, this nonlinear artificial-boundary setting. The problem asks for schemes whose boundary-history evaluation keeps per-step work and memory bounded independently of the accumulated history, with full stability and convergence theory; exact boundary-condition construction beyond separable geometries is a related open direction noted in the same source context.

Background

Heat and mass transfer on unbounded spatial domains is modeled by parabolic equations such as the heat equation, sometimes with nonlinear conditions on a finite boundary - for example a nonlinear flux law −ux=f(u)-u_x=f(u) or a dynamical (time-dependent) boundary condition ut−ux=upu_t-u_x=u^p. Direct numerical solution requires truncating the unbounded domain at an artificial boundary and imposing an artificial boundary condition (ABC) there; an exact ABC reproduces the exterior solution on the truncated computational domain and introduces no modeling error. For the heat equation on a half-line, a Laplace transform in time of the exterior problem and elimination of the exterior unknowns yields a Dirichlet-to-Neumann map, i.e. a boundary operator expressing the normal derivative at the artificial boundary in terms of the history of the boundary trace:

νx(0,t)=−1π ∂∂t∫0tν(0,λ)t−λ dλ,\nu_x(0,t)=-\frac{1}{\sqrt{\pi}}\,\frac{\partial}{\partial t}\int_0^t\frac{\nu(0,\lambda)}{\sqrt{t-\lambda}}\,d\lambda,

which is a half-order fractional (Riemann-Liouville/Caputo-type) derivative of the trace (a fractional derivative is a weighted integral over the past that generalizes integer-order differentiation). On a semi-infinite strip, a modal decomposition in the transverse direction replaces the single kernel by a sum of terms with kernels e−αn(t−λ)/t−λe^{-\alpha_n(t-\lambda)}/\sqrt{t-\lambda}. Koleva and Vulkov (2007) combined these exact ABCs with a finite-element semi-discretization in space and implicit-explicit (IMEX) Euler time stepping - treating the stiff diffusion term implicitly and the nonlinear boundary and memory terms explicitly - proved solvability and stability for the fully discrete 2D scheme with an O(τ3/2)O(\tau^{3/2}) quadrature error, and reproduced analytically known blow-up phenomena (finite-time growth of the solution for supercritical nonlinearities, critical exponents, blow-up sets) numerically.

Exactness has a price: the boundary condition is nonlocal in time. Direct evaluation at step nn reprocesses the whole boundary history, costing O(N2)O(N^2) work and O(N)O(N) memory over NN steps, and the kernel singularity at λ=t\lambda=t obstructs naive recursions. Koleva and Vulkov state that they cannot find time-recursive (local-in-time) update formulas for the singular history integrals J1(t)=∫0tH(t−λ)UAB(λ) dλJ_1(t)=\int_0^t H(t-\lambda)U_{AB}(\lambda)\,d\lambda and J2(t)=∫0tH(t−λ)U˙AB(λ) dλJ_2(t)=\int_0^t H(t-\lambda)\dot U_{AB}(\lambda)\,d\lambda (with UABU_{AB} the trace at the artificial boundary, H(t−λ)=1/t−λH(t-\lambda)=1/\sqrt{t-\lambda} in 1D and H(t−λ)=e−α(t−λ)/t−λH(t-\lambda)=e^{-\alpha(t-\lambda)}/\sqrt{t-\lambda} per strip mode in 2D) and leave this open, together with provably stable higher-order and adaptive time discretizations, efficient algorithms for the long-time slowly growing blow-up regime, and exact ABC construction beyond the semi-infinite line and strip.

The kernel-level obstacle has since been removed by general-purpose machinery, though not for this setting. Convolution quadrature (CQ), introduced by Lubich, discretizes temporal convolutions through a time-integration scheme applied to the Laplace-domain representation of the kernel; Schädle, López-Fernández and Lubich (2006) made CQ fast and oblivious - O(Nlog⁡N)O(N\log N) work and O(log⁡N)O(\log N) active memory, using only evaluations of the Laplace transform of the kernel, assumed sectorial - with applicability to nonlinear convolution equations and a subdiffusion transparent-boundary example; the same authors (2008) gave a variable-step, adaptive version illustrated on a blow-up problem. Complementarily, the sum-of-exponentials (SOE) method of Jiang, Zhang, Zhang and Zhang (2017) approximates the weakly singular power-law kernel by O(log⁡N)O(\log N) exponentials so the history advances by simple recursive updates, with stability and error analysis; for derivative order 1/21/2 this is exactly the 1D kernel above. Fast convolution for nonlocal transparent boundary conditions was demonstrated earlier for the Schrödinger equation by Arnold, Ehrhardt and Sofronov (2003). For the linear heat equation, Wu and Zhang (2011) constructed high-order local absorbing (approximate, non-exact) boundary conditions that avoid the history entirely. The monograph of Han and Wu (2013) records that nonlinear and exact ABC construction remains problem- and geometry-specific. No published work couples the fast-history machinery to the nonlinear boundary dynamics of the Koleva-Vulkov setting and proves stability and convergence of the resulting fully discrete scheme, treats high-order and adaptive stepping near blow-up, or extends exact ABC construction to non-separable geometries.

Problem Statement

Construct computationally efficient and provably stable fully discrete schemes for the heat equation on unbounded domains with nonlinear boundary conditions - such as nonlinear flux −ux=f(u)-u_x=f(u) or dynamical boundary conditions ut−ux=upu_t-u_x=u^p at a finite boundary - truncated by exact artificial boundary conditions, on the semi-infinite line and the semi-infinite strip. The scheme should achieve:

  1. Local-in-time or bounded-memory evaluation of the singular boundary-history convolutions J1(t)=∫0tH(t−λ)UAB(λ) dλJ_1(t)=\int_0^t H(t-\lambda)U_{AB}(\lambda)\,d\lambda and J2(t)=∫0tH(t−λ)U˙AB(λ) dλJ_2(t)=\int_0^t H(t-\lambda)\dot U_{AB}(\lambda)\,d\lambda (with UABU_{AB} the unknown trace of the solution at the artificial boundary and U˙AB\dot U_{AB} its time derivative), with kernels H(t−λ)=1/t−λH(t-\lambda)=1/\sqrt{t-\lambda} (1D) and H(t−λ)=e−α(t−λ)/t−λH(t-\lambda)=e^{-\alpha(t-\lambda)}/\sqrt{t-\lambda} (per strip mode, 2D), such that neither per-step work nor memory grows proportionally to the accumulated history, with a rigorous error bound for the history evaluation as used inside the scheme;
  2. The following are jointly required: (a) high-order accuracy in time, of order at least two; (b) support for adaptive variable time steps; (c) a proven stability and convergence theorem for the coupled nonlinear scheme - including in the long-time, slowly growing blow-up regime studied in the source, where naive exact-ABC implementations become prohibitively expensive.

A further derivation-side direction discussed in the same source context is the extension of exact artificial boundary conditions for heat equations with nonlinear boundary conditions beyond the semi-infinite line and strip to more general geometries and broader classes of nonlinear boundary conditions; it remains open, but is a separate construction problem rather than a required component of the schemes sought here.

The common objective is rigorously justified, computationally feasible exact (modeling-error-free) domain truncation for heat problems with nonlinear boundary dynamics on unbounded domains. Approximate alternatives (local absorbing conditions, perfectly matched layers, or remote simple boundaries) answer a different question and do not by themselves resolve this one.

References

  1. Miglena Koleva and Lubin Vulkov, "Numerical solution of the heat equation with nonlinear boundary conditions in unbounded domains", Numerical Methods for Partial Differential Equations 23(2):379-399, 2007. DOI: 10.1002/num.20183, https://doi.org/10.1002/num.20183
  2. Achim Schädle, María López-Fernández and Christian Lubich, "Fast and Oblivious Convolution Quadrature", SIAM Journal on Scientific Computing 28(2):421-438, 2006. DOI: 10.1137/050623139, https://doi.org/10.1137/050623139
  3. María López-Fernández, Christian Lubich and Achim Schädle, "Adaptive, Fast, and Oblivious Convolution in Evolution Equations with Memory", SIAM Journal on Scientific Computing 30(2):1015-1037, 2008. DOI: 10.1137/060674168, https://doi.org/10.1137/060674168
  4. Shidong Jiang, Jiwei Zhang, Qian Zhang and Zhimin Zhang, "Fast Evaluation of the Caputo Fractional Derivative and Its Applications to Fractional Diffusion Equations", Communications in Computational Physics 21(3):650-678, 2017. DOI: 10.4208/cicp.oa-2016-0136, https://doi.org/10.4208/cicp.oa-2016-0136
  5. Yonggui Yan, Zhi-Zhong Sun and Jiwei Zhang, "Fast Evaluation of the Caputo Fractional Derivative and Its Applications to Fractional Diffusion Equations: A Second-Order Scheme", Communications in Computational Physics 22(4):1028-1048, 2017. DOI: 10.4208/cicp.oa-2017-0019, https://doi.org/10.4208/cicp.oa-2017-0019
  6. Anton Arnold, Matthias Ehrhardt and Ivan Sofronov, "Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability", Communications in Mathematical Sciences 1(3):501-556, 2003. DOI: 10.4310/cms.2003.v1.n3.a7, https://doi.org/10.4310/cms.2003.v1.n3.a7
  7. Xiaonan Wu and Jiwei Zhang, "High-Order Local Absorbing Boundary Conditions for Heat Equation in Unbounded Domains", Journal of Computational Mathematics 29(1):74-90, 2011. DOI: 10.4208/jcm.1004-m3195, https://doi.org/10.4208/jcm.1004-m3195
  8. Houde Han and Xiaonan Wu, "Artificial Boundary Method", Springer, Berlin, 2013. DOI: 10.1007/978-3-642-35464-9, https://doi.org/10.1007/978-3-642-35464-9
  9. Miglena N. Koleva, "Numerical Solution of the Heat Equation in Unbounded Domains Using Quasi-uniform Grids", Lecture Notes in Computer Science, pp. 509-517, 2006. DOI: 10.1007/11666806_58, https://doi.org/10.1007/11666806_58