Adaptive multiresolution approach for solution of hyperbolic PDEs

Abstract

This paper establishes an innovative and efficient multiresolution adaptive approach combined with high-resolution methods, for the numerical solution of a single or a system of partial differential equations. The proposed methodology is unconditionally bounded (even for hyperbolic equations) and dynamically adapts the grid so that higher spatial resolution is automatically allocated to domain regions where strong gradients are observed, thus possessing the two desired properties of a numerical approach: stability and accuracy.Numerical results for five test problems are presented which clearly show the robustness and cost effectiveness of the proposed method.

Adaptive multiresolution mesh refinement for the solution of evolution PDEs

“…In this case, we choose x 3,0 in order to equalize the number of points on both sides. Hence, our final set X near consists of points x 3,0 , x 3,2 , x 3,5 , and x 3,6 as shown in Figure 2.…”

“…Subsequently we check y 3,2 . Since g(y 3,2 ) cannot be interpolated from the nearest two points y 2,0 , y 2,1 ∈ Grid int we include y 3,2 along with points y 3,1 and y 3,3 (which in any case is already present in Grid int ) in Grid int . Moving on to the next point y 3,3 , we see again that g(y 3,3 ) cannot be interpolated from the nearest two points y 2,1 , y 2,2 ∈ Grid int .…”

“…Moving on to the next point y 3,3 , we see again that g(y 3,3 ) cannot be interpolated from the nearest two points y 2,1 , y 2,2 ∈ Grid int . Hence, we include y 3,3 along with y 3,2 (which in any case is already present in Grid int ) and y 3,4 in Grid int . The next point in W 3 is y 3,4 .…”

“…A nice survey on some of the earlier works can be found in [5,43]. Adaptive methods currently popular for solving PDEs are: (i) moving mesh methods [1,2,4,6,7,30,13,17,35,36] where one explicitly derives an equation governing the grid, which moves a grid of a fixed number of finite difference cells or finite elements so as to follow and resolve local irregularities in the solution, (ii) the so called "Adaptive Mesh Refinement" method [8,9,10,11] in which the mesh is refined locally based on the difference between the solutions computed on the coarser and the finer grids, and (iii) wavelet-based or multiresolution-based methods [3,12,20,21,24,25,27,44,45,46] which take advantage of the fact that functions with localized regions of sharp transition can be very well compressed. Our proposed method falls under this last category.…”

“…Recently, Alves et al [3] proposed an adaptive multiresolution mesh refinement scheme, similar to the multiresolution mesh refinement approach proposed by Harten 1 [20,21] and the interpolating wavelet approach proposed by Bertoluzza [12] and Holmstrom [24]. The authors in [20,21,24] used the interpolating subdivision scheme of [15,16,20] to construct the interpolating polynomials, whereas Alves et al [3] used the high-resolution schemes SMART [18] and MINMOD [19] for constructing the interpolating polynomials. These approaches share similar underlying ideas.…”