High resolution methods for time dependent problems with piecewise smooth solutions

TL;DR

This work addresses the challenge of solving nonlinear time-dependent convection-dominated problems that spontaneously form shocks and other non-smooth features. It unifies two complementary approaches: local high-resolution central schemes that evolve piecewise smooth approximations without relying on Riemann solvers, and global spectral-like methods augmented with adaptive edge-detection and spectral viscosity to preserve accuracy in smooth regions while controlling Gibbs phenomena. The central schemes (e.g., Nessyahu-Tadmor, Kurganov-Tadmor) provide robust, non-oscillatory solutions via nonlinear adaptivity to the direction of smoothness and wave propagation, while adaptive spectral viscosity and Gelb-Tadmor-type edge detection enable high-resolution global projections to maintain spectral accuracy in smooth regions and recover first- or higher-order convergence away from discontinuities. The hybrid toolkit—comprising nonlinear limiters, adaptive stencils, and entropy-stable spectral stabilization—offers practical, black-box solvers for convection-dominated PDEs across structured and unstructured grids, with demonstrated effectiveness in 1D and multidimensional settings and applications including Burgers-type problems and MHD flows.

Abstract

A trademark of nonlinear, time-dependent, convection-dominated problems is the spontaneous formation of non-smooth macro-scale features, like shock discontinuities and non-differentiable kinks, which pose a challenge for high-resolution computations. We overview recent developments of modern computational methods for the approximate solution of such problems. In these computations, one seeks piecewise smooth solutions which are realized by finite dimensional projections. Computational methods in this context can be classified into two main categories, of local and global methods. Local methods are expressed in terms of point-values (-- Hamilton-Jacobi equations), cell averages (-- nonlinear conservation laws), or higher localized moments. Global methods are expressed in terms of global basis functions. High resolution central schemes will be discussed as a prototype example for local methods. The family of central schemes offers high-resolution ``black-box-solvers'' to an impressive range of such nonlinear problems. The main ingredients here are detection of spurious extreme values, non-oscillatory reconstruction in the directions of smoothness, numerical dissipation and quadrature rules. Adaptive spectral viscosity will be discussed as an example for high-resolution global methods. The main ingredients here are detection of edges in spectral data, separation of scales, adaptive reconstruction, and spectral viscosity.

Figures (8)

• Figure 1: Second-order central scheme simulation of semiconductor device governed by 1D Euler-Poisson equations. Electron velocity in $10^7$ cm/s with $N=400$ cells.
• Figure 2: Third-order central scheme simulation of 1D MHD Riemann problem. with $N=400$ cells. The $y$-magnetic field at $t=0.2$.
• Figure 3: Convection-diffusion eq. $u_t + (u^2)_x/2= (u_x/\sqrt{1+u_x^2})_x$ simulated by (\ref{['eq:semi_discrete_flux']}),(\ref{['eq:minmod']}), with $400$ cells.
• Figure 4: Third-order central scheme for 2D Riemann problem. Density contour lines with $400\times 400$ cells.
• Figure 5: Adaptive reconstruction of the piecewise smooth data from its using $N=128$-modes using (\ref{['eq:adaptive_mollifier']}).