An Alternating-Direction Implicit Orthogonal Spline Collocation Scheme for Nonlinear Parabolic Problems on Rectangular Polygons

Abstract

We consider a nonlinear parabolic initial-boundary value problem on a rectangular polygon with the solution satisfying Robin boundary conditions with variable coefficients. An approximation to the solution at the desired time value is obtained using an alternating-direction implicit extrapolated Crank-Nicolson scheme in which orthogonal spline collocation with piecewise polynomials of an arbitrary degree is used for spatial discretization. At each time level, the scheme determines the intermediate solution along horizontal lines and the approximate solution along vertical lines passing through the Gauss points. Only at the last time level is the approximate solution along vertical lines converted into the approximate solution defined on the entire rectangular polygon. This property of our approach leads to its efficient implementation and its applicability to rectangular polygons.

An ADI extrapolated Crank–Nicolson orthogonal spline collocation method for nonlinear reaction–diffusion systems on evolving domains

“…2 Taking these matters into account, the parameters in (5.1)-(5.5) are γ = 1, ρ(t) = 1 + 9 sin(πt/1000), t ∈ [0, 1000],…”

“…Based on theoretical analyses of linear parabolic problems [1] and numerical results for nonlinear parabolic equations and systems [2,8], heuristically we expect the ADI scheme (Steps 1 to 4) to yield the approximation u h ∈ M × M to u(·, T ) on ∪ ∂ such that u(·, T ) − u h L 2 ( ) ≤ C 1 (τ 2 + h r+1 ), (4.22) and u(·, T ) − u h L ∞ ( ) ≤ C 1 (τ 2 + h r+1 ), (4.23) where the positive constant C 1 is independent of h and τ , for a regular collection of partitions of .…”

“…In this paper, using [2], we generalize the ADI OSC method in [8,9] to solve problems that arise when the region is not fixed but evolves in time. The initial-boundary value problem modeling this behavior takes the form (see [10][11][12][13][14][15][16][17][18][19]26]) ∂u ∂t − D u + (∇ · a)u + (a · ∇)u = f(u), (x, y, t) ∈ t × [0, T ], (1.4) n 1 ∂u ∂x + n 2 ∂u ∂ y = 0, (x, y, t) ∈ ∂ t × (0, T ], (1.5) u(x, y, 0) = g(x, y), (x, y) ∈ 0 ∪ ∂ 0 , (1.6) where ∇ ≡ [∂/∂x, ∂/∂ y] T is the gradient operator, a = [a 1 (x, y, t), a 2 (x, y, t)] T is a given vector field, · denotes the standard inner product in R 2 , t ⊂ R 2 is a time-dependent domain with boundary ∂ t and n 1 (x, y, t), n 2 (x, y, t) are the elements of an outward/inward normal vector on the boundary ∂ t .…”

“…We note that if L 1 , L 2 are nonlinear, that is, they are functions of u, ∂u ∂x , ∂u ∂ y , the ADI scheme can be generalized based on the ADI scheme for a single parabolic equation in[2].…”

High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements

“…(In fact, the previous Fourier approaches exhibit even less flexibility than their finite-difference counterparts: the latter are applicable to general domains, albeit with first-order accuracy.) Spatial spline collocation methods have also been introduced in this context, see [51][52][53] and references therein. In particular, cubic-spline interpolation was used [53] in conjunction with an alternating direction embedding scheme to solve elliptic PDEs for non-rectangular geometries, but only conditional stability was obtained (for the Heat-Equation-like discrete scheme whose steady state provides the solution of the given elliptic equation).…”