We study discontinuous Galerkin (DG) discretizations — combining the mathematical rigor of finite element methods with the local conservation of finite volume schemes — for hyperbolic conservation laws whose solutions develop shocks and rarefaction waves in finite time. From Runge–Kutta DG, numerical flux design and limiting to entropy-stable and Lagrange-multiplier (DGLM) formulations, our work spans rigorous analysis, implementation, and application.
Jaemin Shin · Principal Investigator · ✉ jmshin@yonsei.ac.kr
The Yonsei DG Lab is an applied and computational mathematics research group operating on the Yonsei University campus. Hyperbolic conservation laws \(u_t+\nabla\cdot f(u)=0\) encompass core models of nature and engineering — the Euler equations of gas dynamics, the shallow water equations, traffic flow — yet even perfectly smooth initial data develop shock waves in finite time, so a discretization must deliver high-order accuracy and nonlinear stability at once. We study one of the most successful answers to this challenge, the discontinuous Galerkin method, from many angles: element-wise high-order polynomial approximation, Riemann-solver-based numerical fluxes, limiters that control spurious oscillations, discrete entropy inequalities, and new formulations built on Lagrange multipliers.
An applied mathematician by training, he specializes in finite element and discontinuous Galerkin methods for partial differential equations. Alongside high-order discretizations of conservation laws, he applies SINDy — sparse identification of governing differential equations from data — to storage-reliability telemetry (HDD SMART attributes) for real-time failure prediction.
The classical Cockburn–Shu framework: weak formulations on piecewise-polynomial spaces combined with strong-stability-preserving (SSP) Runge–Kutta time integration. We analyze how TVD/TVB control of cell averages coexists with genuine high-order accuracy.
High-orderSSP-RKGodunov, Roe, HLL/HLLC and (local) Lax–Friedrichs fluxes examined through consistency, monotonicity and upwinding — quantifying the robustness–resolution trade-off across problem classes.
Riemann solversUpwindingminmod/TVB and WENO-based limiters, plus Zhang–Shu scaling for maximum-principle and positivity preservation (density, pressure) — taming Gibbs oscillations near shocks without sacrificing formal order of accuracy.
Shock capturingBound-preservingDiscrete entropy inequalities built from entropy pairs \((\eta,q)\): Tadmor's entropy-conservative two-point fluxes, flux differencing, and summation-by-parts (SBP) operators yielding provably nonlinearly stable high-order schemes.
Entropy stabilitySBPInterface continuity and flux conditions imposed weakly through Lagrange multipliers on the mesh skeleton: saddle-point structure, inf-sup stability, static condensation, and the extension to time-dependent conservation laws.
Lagrange multiplierHybridizationBurgers, Euler and shallow-water simulations with GPU parallelism — and, beyond PDEs, SINDy-based discovery of degradation dynamics from HDD SMART telemetry for real-time disk-failure prediction.
Euler / SWESINDyReliabilityThe domain is partitioned into elements \(K\), and on each element the solution is approximated in the polynomial space \(\mathcal{P}^k(K)\). No continuity is enforced across elements — coupling happens only through the numerical flux \(\widehat{f}\) on element boundaries.
Time integration uses SSP Runge–Kutta methods, which preserve the stability of the spatial discretization (third-order example):
The CFL condition scales like \(\Delta t \lesssim \Delta x /\big[(2k+1)\,\max|f'(u)|\big]\), tightening with the polynomial degree \(k\).
Consistent, monotone numerical fluxes are the key to stability and to selecting the entropy solution. The simplest choice is the (local) Lax–Friedrichs flux:
The Godunov flux uses the exact solution of local Riemann problems and carries the least numerical viscosity; for systems, Roe and HLL/HLLC approximate Riemann solvers balance accuracy against cost. Both Lax–Friedrichs and Godunov can be compared side by side in the live demo below.
Gibbs oscillations near shocks are suppressed by minmod-based slope limiting, which compares edge deviations against neighboring differences of cell means:
Accuracy loss at smooth extrema is mitigated by the TVB correction (threshold constant \(M\)). The Zhang–Shu scaling limiter instead shrinks the high-order polynomial toward its cell average, preserving maximum principles and positivity without destroying the order of accuracy:
\([m,M]\) is the global range to be preserved; \(m_K, M_K\) are the extrema of the polynomial on cell \(K\).
Physically admissible weak solutions satisfy an entropy inequality for every convex entropy pair \((\eta,q)\). With entropy variables \(v=\eta'(u)\) and the entropy potential \(\psi=v\cdot f(u)-q(u)\), two-point fluxes \(f_S\) satisfying Tadmor's condition — combined in flux-differencing form — yield high-order DG schemes that are provably entropy stable at the discrete level:
Imposing the interface conditions (trace continuity, flux matching) weakly through a Lagrange multiplier \(\lambda_h\) defined on the mesh skeleton turns the scheme into a mixed formulation with saddle-point structure:
Inf-sup stability of the multiplier space, element-local elimination by static condensation, and the connection to hybridizable DG are central themes — and extending this framework to time-dependent hyperbolic conservation laws is a flagship direction of the lab.
The 1D Burgers equation \(u_t+\big(u^2/2\big)_x=0\) with initial data \(u_0(x)=\tfrac12+\sin 2\pi x\) and periodic boundaries, computed live in your browser (\(P^1\) RKDG, SSP-RK2, limiting after every stage). Each cell carries its own linear polynomial, so the jumps at element interfaces (gold vertical lines) are plainly visible — the signature of DG. Watch a shock form at \(t\approx0.16\) and remain oscillation-free under limiting; switch the limiter off to see Gibbs oscillations appear near the shock.
How the TVB threshold constant M trades accuracy at smooth extrema against oscillation control near shocks, benchmarked on scalar conservation laws — with a scaling rule for M under mesh refinement.
From Lax–Friedrichs to Godunov: shock resolution, numerical viscosity and convergence order under an identical P² RKDG discretization — including an entropy-violation case at a sonic point inside a rarefaction wave.
Working out Tadmor's condition \((v_R-v_L)^\top f_S=\psi_R-\psi_L\) for the Burgers and shallow-water equations, and coupling the resulting fluxes into a flux-differencing DG scheme, step by step.
Applying weak-form SINDy to HDD SMART telemetry to identify degradation dynamics without computing derivatives — and how test-function design borrowed from conservation-law weak forms improves robustness to noise.
