The artificial boundary method (ABM) of Wu and Zhang (2008) reduces the Cauchy problem for the two-dimensional vector Burgers equation on the plane, with compactly supported data and irrotational flow outside the data support, to an initial-boundary value problem on a disc: a Cole–Hopf transformation linearizes the exterior to the heat equation, whose exact circular Dirichlet-to-Neumann relation is an infinite angular Fourier series with slowly decaying temporal convolution kernels, truncated at M modes and then discretized by a Crank–Nicolson finite-difference scheme. The method is formally derived and numerically validated, but it has never been given a rigorous foundation: no bound quantifies the truncation error in M or the quadrature error of the history integrals; the a priori stability bound holds only under structural hypotheses that are not verifiable for the reduced or discrete problem; and no proof links the discrete scheme to the solution of the original whole-plane problem. The corresponding analysis program has been completed only in one space dimension and for linear equations or local (Padé-type) boundary conditions, while the two-dimensional nonlinear nonlocal reduction itself remains without stability, error, or convergence theory. The problem posed here is to establish that theory; a provably accurate fast evaluation of the history convolution kernels is a complementary algorithmic direction.

Background

The viscous Burgers equation is the classical nonlinear model combining advection and diffusion, and a standard testbed for numerical methods aimed at turbulence-type flows. In two dimensions its coupled (vector) form is

ut=∇2u+2(u⋅∇)u+F(x,t),u=(u1,u2),\mathbf{u}_t=\nabla^2\mathbf{u}+2(\mathbf{u}\cdot\nabla)\mathbf{u}+\mathbf{F}(x,t),\qquad \mathbf{u}=(u_1,u_2),

posed on the whole plane R2\mathbb{R}^2 with initial data and source compactly supported inside a disc of radius RR, and with u→0\mathbf{u}\to0 at infinity. Direct discretization is impossible on an unbounded domain, so one introduces a circular artificial boundary r=Rr=R and seeks boundary conditions on it that reproduce, exactly or to controlled accuracy, the effect of the discarded exterior region; this is the artificial boundary method (ABM). The ideal boundary relation is a Dirichlet-to-Neumann (DtN) map, an operator that takes the history of boundary values and returns the boundary flux; for linear exterior equations such maps are explicit, and for the heat equation the ABM is on firm ground: exact and approximating parabolic boundary conditions were derived by Han and Huang (2002), and Wu and Sun (2004) proved convergence of a difference scheme for the heat equation on unbounded domains using such conditions. For nonlinear equations the exact exterior DtN map is not available in general. The key idea of Wu and Zhang (2008), extending the one-dimensional treatment of Han, Wu, and Xu (2006), is that if the flow is irrotational (has zero vorticity ∇×u\nabla\times\mathbf{u}) in the exterior region where the data vanish, then a Cole–Hopf transformation — the classical substitution that linearizes Burgers' equation, here a potential ω\omega with eωui=∂xiv\mathrm{e}^{\omega}u_i=\partial_{x_i}v for a scalar vv — turns the exterior problem into the linear heat equation for vv. The heat-equation DtN map on the circle r=Rr=R is an infinite Fourier series in the angular variable whose coefficients are temporal convolutions with kernels HmH_m defined by integrals involving the squares of the Bessel functions JmJ_m and YmY_m; these kernels decay slowly, which makes both their analysis and their evaluation expensive. Inverting the transformation yields the exact boundary condition for u\mathbf{u} as a nonlinear, history-dependent functional, and truncating the angular series at MM modes gives a family of computable approximating boundary operators, which close a bounded-domain initial-boundary value problem on the disc. Wu and Zhang established an a priori L2L^2 stability bound for the reduced problem (their Theorem 1), growing in time at most like e(2C+3)t\mathrm{e}^{(2C+3)t} in terms of the data — but under structural hypotheses: a uniform bound ∣u1u2∣≤C|u_1u_2|\le C and square-integrability of u\mathbf{u} over all of R2\mathbb{R}^2, conditions that encode the exterior problem and are not verified for the solution of the reduced problem itself. They then discretized the reduced problem by a Crank–Nicolson scheme (the standard second-order implicit time-stepping method) with fixed-point linearization of the convective terms, trapezoidal quadrature of the boundary history integrals, and precomputed kernel integrals, and validated the scheme on manufactured solutions, observing errors that decrease under mesh refinement and as MM grows. What the paper does not contain is any rigorous truncation analysis: no estimate of the error committed by replacing the infinite angular series by its MM-mode truncation, no error bound for the temporal quadrature of the history integrals, and no convergence theorem connecting the fully discrete solution to the solution of the original whole-plane problem. Subsequent literature makes the gap stand out. In one space dimension the parallel program is essentially complete: Sun and Wu (2009) proved solvability, stability, and convergence of order O(τ3/2+h2)O(\tau^{3/2}+h^2) for a Burgers difference scheme on a finite interval with exact artificial boundary conditions, and Zheng, Zhao, and Liu (2017) constructed a linearized tridiagonal scheme with full stability and convergence analysis, leaving only faster convolution evaluation and higher-order boundary conditions open in 1D. Rigorous analyses also exist for semilinear parabolic equations with nonlinear local (Padé-approximation-based) absorbing boundary conditions (Sun, Wu, Zhang, and Wang, 2012), for one-dimensional time-fractional nonlinear parabolic and Burgers problems (Li and Zhang, 2016; Li and Wu, 2021), and for fast convolution evaluation of artificial boundary conditions for one-dimensional advection-diffusion equations (Sun, Wang, and Zheng, 2020), whose authors state the multidimensional extension as open. For the two-dimensional Burgers equation itself, the direct successors of Wu and Zhang are alternative computational methods — the Fourier-space integral-equation approaches of Egidi and Maponi (2016) and Egidi, Maponi, and Quadrini (2018) — which likewise come without a rigorous stability or convergence analysis; local nonlinear boundary conditions on a circle were used for semilinear heat equations by Zhang, Han, and Brunner (2011), also without reduction error analysis. The monograph of Han and Wu (2013) surveys nonlinear artificial boundary conditions for Burgers-type equations and reports stability results, but no two-dimensional convergence theory. Thus the exact, nonlocal, two-dimensional nonlinear reduction of Wu and Zhang (2008) remains the flagship case of the ABM program without rigorous error control.

Problem Statement

Consider the Cauchy problem for the two-dimensional vector Burgers equation ut=∇2u+2(u⋅∇)u+F\mathbf{u}_t=\nabla^2\mathbf{u}+2(\mathbf{u}\cdot\nabla)\mathbf{u}+\mathbf{F} on R2\mathbb{R}^2 with data compactly supported inside the disc r<Rr<R and with irrotational flow (∇×u=0\nabla\times\mathbf{u}=0) in the exterior, and the Wu–Zhang artificial boundary reduction of it: the initial-boundary value problem on the disc r≤Rr\le R closed by the truncated nonlinear nonlocal boundary operators κ1M,κ2M\kappa_{1M},\kappa_{2M} (angular Fourier truncation at MM modes of the exact exterior Cole–Hopf/heat Dirichlet-to-Neumann relation with convolution kernels HmH_m), together with its Crank–Nicolson finite-difference discretization using trapezoidal quadrature of the boundary history integrals. Establish a rigorous foundation for this reduction and its discretization, namely: (a) well-posedness and an a priori stability bound for the reduced problem with the truncated operators under hypotheses verifiable from the problem data, replacing the structural assumptions that ∣u1u2∣|u_1u_2| is uniformly bounded and that u∈L2(R2)\mathbf{u}\in L^2(\mathbb{R}^2); (b) explicit bounds on the boundary-operator error as a function of the truncation parameter MM, and on the error of the temporal quadrature of the history integrals; (c) convergence of the fully discrete scheme to the solution of the original whole-plane problem, with convergence rates in the radial step, the angular step, the time step, and MM. These components jointly serve one scientific objective — placing the two-dimensional Burgers artificial boundary method on the same rigorous footing already achieved for its one-dimensional analogues and for linear equations — and should be understood as linked parts of a single analysis program for the same reduced problem and the same discretization, not as independent problems. The objective comprises the three interlinked results (a)–(c). Certified fast evaluation of the HmH_m history convolutions (for example by compressed or rational representations), achieved so far only for one-dimensional linear symbols, is a complementary algorithmic direction.

References

  1. Xiaonan Wu and Jiwei Zhang, 'Artificial boundary method for two-dimensional Burgers' equation', Computers & Mathematics with Applications 56(1) (2008) 242–256. DOI: 10.1016/j.camwa.2007.11.047, https://doi.org/10.1016/j.camwa.2007.11.047. (zbMATH: Zbl 1145.65321, https://zbmath.org/5355543.)
  2. Houde Han and Zhongyi Huang, 'Exact and approximating boundary conditions for the parabolic problems on unbounded domains', Computers & Mathematics with Applications 44(5–6) (2002) 655–666. DOI: 10.1016/S0898-1221(02)00180-3, https://doi.org/10.1016/S0898-1221(02)00180-3.
  3. Xiaonan Wu and Zhi-Zhong Sun, 'Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions', Applied Numerical Mathematics 50 (2004) 261–277. DOI: 10.1016/j.apnum.2004.01.001, https://doi.org/10.1016/j.apnum.2004.01.001.
  4. Houde Han, Xiao-nan Wu, and Zhenli Xu, 'Artificial boundary methods for Burgers' equation using nonlinear boundary conditions', Journal of Computational Mathematics 24(3) (2006) 295–304. zbMATH: 5038551 (Zbl 1096.65085), https://zbmath.org/5038551.
  5. Zhi-Zhong Sun and Xiao-Nan Wu, 'A difference scheme for Burgers equation in an unbounded domain', Applied Mathematics and Computation 209(2) (2009) 285–304. DOI: 10.1016/j.amc.2008.12.052, https://doi.org/10.1016/j.amc.2008.12.052.
  6. Quan Zheng, Xin Zhao, and Yufeng Liu, 'A novel finite difference scheme for Burgers' equation on unbounded domains', Applied Numerical Mathematics 111 (2017) 1–16. DOI: 10.1016/j.apnum.2016.09.002, https://doi.org/10.1016/j.apnum.2016.09.002.
  7. Nadaniela Egidi and Pierluigi Maponi, 'Artificial boundary conditions for the Burgers equation on the plane', Applied Mathematics and Computation 286 (2016) 1–14. DOI: 10.1016/j.amc.2016.04.008, https://doi.org/10.1016/j.amc.2016.04.008.
  8. Nadaniela Egidi, Pierluigi Maponi, and Michela Quadrini, 'An integral equation method for the numerical solution of the Burgers equation', Computers & Mathematics with Applications 76(1) (2018) 35–44. DOI: 10.1016/j.camwa.2018.04.002, https://doi.org/10.1016/j.camwa.2018.04.002.
  9. Zhi-zhong Sun, Xiaonan Wu, Jiwei Zhang, and Desheng Wang, 'A linearized difference scheme for semilinear parabolic equations with nonlinear absorbing boundary conditions', Applied Mathematics and Computation 218 (2012) 5187–5201. DOI: 10.1016/j.amc.2011.10.083, https://doi.org/10.1016/j.amc.2011.10.083.
  10. Jiwei Zhang, Houde Han, and Hermann Brunner, 'Numerical Blow-up of Semilinear Parabolic PDEs on Unbounded Domains in ℝ²', Journal of Scientific Computing 49 (2011) 367–382. DOI: 10.1007/s10915-011-9467-5, https://doi.org/10.1007/s10915-011-9467-5.
  11. Dongfang Li and 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.
  12. Hongwei Li and Yuchen Wu, 'Artificial boundary conditions for nonlinear time fractional Burgers' equation on unbounded domains', Applied Mathematics Letters 120 (2021) 107277. DOI: 10.1016/j.aml.2021.107277, https://doi.org/10.1016/j.aml.2021.107277.
  13. Ting Sun, Jilu Wang, and Chunxiong Zheng, 'Fast Evaluation of Artificial Boundary Conditions for Advection Diffusion Equations', SIAM Journal on Numerical Analysis 58(6) (2020) 3530–3557. DOI: 10.1137/19M130145X, https://doi.org/10.1137/19M130145X.
  14. Xinyue Chen, Wenqing Zhang, and Yuchen Wu, 'Discrete Transparent Boundary Conditions for Burgers' Equation', Engineering 16 (2024) 423–437. DOI: 10.4236/eng.2024.1612031, https://doi.org/10.4236/eng.2024.1612031.
  15. Houde Han and Xiaonan Wu, 'Nonlinear Artificial Boundary Conditions', Chapter 9 of 'Artificial Boundary Method', Springer Berlin Heidelberg (2013) 341–363. DOI: 10.1007/978-3-642-35464-9_9, https://doi.org/10.1007/978-3-642-35464-9_9.