Essentially Non-oscillatory Spectral Volume Methods

TL;DR

The paper addresses high-order, non-oscillatory reconstruction for 1D hyperbolic conservation laws by recasting ENO recovery as a nonlinear variational problem constrained to a convex subset. A subset-selection strategy together with convex projections and an active-set/conjugate-gradient solver yields a nonlinear, fixed-stencil spectral-volume recovery that enforces a sign property and limits oscillations near shocks. Practical implementation uses Legendre bases, a linear recovery operator $\mathcal{R}=V\mathcal{A}^{-1}$, and jump-type basis functions to represent discontinuities, with Chebyshev node layouts to control conditioning. Numerical tests on static and Euler problems show high-order convergence with oscillations confined to shocked macrocells, and the approach demonstrates potential for efficient, tunable high-order SV reconstructions, including future extension to 2D grids.

Abstract

A new Essentially Non-oscillatory (ENO) recovery algorithm is developed and tested in a Finite Volume method. The construction is hinged on a reformulation of the reconstruction as the solution to a variational problem. The sign property of the classical ENO algorithm is expressed as restrictions on the admissible set of solutions to this variational problem. In conjunction with an educated guessing algorithm for possible locations of discontinuities an ENO reconstruction algorithm without divided differences or smoothness indicators is constructed. No tunable parameters exist apart from the desired order and stencil width. The desired order is in principle arbitrary, but growing stencils are needed. While classical ENO methods consider all connected stencils that surround a cell under consideration the proposed recovery method uses a fixed stencil, simplifying efficient high order implementations.

Figures (5)

• Figure 1: The prototype inter cell jump function $\Psi_{j-\frac{1}{2}}(x)$, centerd at the cell edge between cells $j-1$ and $j$. The support of the function is limited to the cells $j-1$ and $j$. The average values of the function in cell $j-1$ is $-1 \slash 2$, while it is $1 \slash 2$ in cell $j$. The function jumps from $-1$ to $1$ at the edge $x_{j-\frac{1}{2}}$.
• Figure 2: Active set method in conjunction with the conjugate gradient method. The contour lines of the target function are depicted as ellipses, a line separates the admissible half-space from the non-admissible half-space. A blue and a red arrow are a first and second step of the active set method. The blue step can be seen as a direct solution of the unrestricted problem using the CG method. As this step would leave the half-space of admissible solutions its length is restricted to the point where it collides with the hyperplane separating admissible and non-admissible solutions. The following step is constrained to the hyperplane, but no step length restrictions are needed. After two steps, the restricted minimizer is found.
• Figure 3: Initial condition $u_1$ with overlaid recovery. Cell edges are shown as vertical grey lines. The green circles are the right cell edge values, while the red circles give the left edge values of the recovered function. The jump height is recovered nearly exact.
• Figure 4: Recovery of a smooth function $u_2 = \sin(x)$ with overlaid recovery. Cell edges are shown as vertical grey lines. The green circles are the right cell edge values, while the red circles give the left edge values of the recovered function. All function values are recovered with high accuracy. While two jumping basis functions are part of the admissible set of basis functions their contribution is negligible. The right edge value of every cell coincides with the left edge value of the neighbouring cell.
• Figure 5: Recovery of function $u_3$ with smooth nonconstant areas connected by a discontinuity. Cell edges are shown as vertical grey lines. The green circles are the right cell edge values, while the red circles give the left edge values of the recovered function. As before, the jump height is predicted with high accuracy, the smooth parts are recovered with negligible oscillations.