A partial differential equation approach to multidimensional extrapolation

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

“…This is done with extrapolation procedures, following Aslam [8] and described in 4.2. Also, in the case of the design of high-order accurate schemes, it is necessary to guarantee that time evolution procedures are adequate; this will be described in section 4.3.…”

“…The rationale for extrapolations in the normal direction is based on the fact that the interface propagates only in its normal direction 4 . The extrapolation procedures we use are those of [8], detailed in section 4.2. The procedure for solving the Stefan problem follows the algorithm given in algorithm 1. t n := t n+1 , 5.…”

“…However, in the case of defining a valid right-hand-side for equation (9), high-order extrapolations are necessary. Such high-order extrapolations in the normal direction are performed in a series of steps, as proposed in Aslam [8]. For example, suppose that one needs to generate a cubic extrapolation of a scalar quantity Q from the region where φ ≤ 0 to the region where φ > 0.…”

A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem

“…High order extrapolation in the normal direction is performed in a series of steps, as proposed in [1]. For example, suppose that we seek to extrapolate T from the region where φ ≤ 0 to the region where φ > 0.…”

“…Figure 9 illustrates cubic extrapolation. This example is taken from [1]. Consider a computational domain Ω = [−π, π] × [−π, π] separated into two regions: Ω − defined as the interior of a disk with center at the origin and radius two, and its complementary Ω + .…”