Runge-Kutta Discontinuous Galerkin Schemes

• Runge-Kutta Discontinuous Galerkin schemes are high-order finite element methods that solve time-dependent PDEs using discontinuous representations and explicit Runge-Kutta integration.
• They incorporate δ-mapping at discontinuous flux interfaces to unify left/right states, ensuring steady-state preservation and non-oscillatory shock resolution.
• Practical applications include nonlinear elasticity in layered media and multi-class traffic flow, demonstrating entropy consistency and high accuracy.

A Runge-Kutta Discontinuous Galerkin (RKDG) scheme is a high-order finite element method for numerically solving time-dependent partial differential equations, notably hyperbolic conservation laws. The RKDG framework applies a spatial discretization by discontinuous Galerkin (DG) finite elements, combined with strong-stability-preserving (SSP) explicit Runge-Kutta time integration. When the flux function contains spatially discontinuous coefficients—i.e., $u_t + f\big(u, \theta(x)\big)_x = 0$ where $\theta(x)$ is a piecewise constant parameter—the DG scheme requires special treatment at interfaces of discontinuity to retain steady-state preservation, high accuracy, and non-oscillatory shock resolution. The hybrid RKDG scheme augmented with $\delta$-mapping achieves these goals by unifying left/right states at discontinuous interfaces and adjusting limiters to maintain entropy and supply-demand consistency (Qiao et al., 2015).

1. Problem Formulation and Flux Discontinuity

Consider the initial-value problem for a hyperbolic conservation law with spatially variable flux parameters,

$$u_t + \partial_x f(u, \theta(x)) = 0, \qquad x \in [0, L],~t > 0,$$

where $u \in \mathbb{R}^r$ and $\theta(x) \in \mathbb{R}^q$ is piecewise constant, possibly discontinuous at mesh interfaces $x_{j+1/2}$. At such interfaces, denote the left and right limits by $\theta^-(x_{j+1/2}) = \theta_j$ and $\theta^+(x_{j+1/2}) = \theta_{j+1}$, respectively.

Classical steady or stationary-shock solutions across jumps satisfy the local Rankine–Hugoniot condition,

$$[f(u, \theta_j)] = 0 \quad\text{across each jump of }\theta,$$

ensuring conservation is retained in the presence of spatially discontinuous flux.

2. Spatial Discretization via Discontinuous Galerkin

Partition $\theta(x)$0 into cells $\theta(x)$1. On each cell, approximate $\theta(x)$2 by polynomials of degree $\theta(x)$3,

$\theta(x)$4

where $\theta(x)$5 is a local basis (e.g., scaled Legendre polynomials).

The weak DG form, for each basis index $\theta(x)$6, reads

$\theta(x)$7

After expansion, the method yields a finite-dimensional ODE system for each coefficient,

$\theta(x)$8

3. Numerical Flux Definition via δ-Mapping

Direct use of standard numerical fluxes at interfaces where $\theta(x)$9 jumps fails to preserve steady states. Instead, the $\delta$0-mapping algorithm maps left/right DG traces $\delta$1 to unified states $\delta$2 evaluated at an intermediate parameter $\delta$3, defined to satisfy:

• Supply-Demand Equalization: maximize $\delta$4 subject to

$\delta$5

• Entropy and Eigenvalue Compatibility:

$\delta$6

ensuring no sign change for characteristic speeds.

The mapped states are used in a classical Riemann solver,

$\delta$7

4. Slope Limiting with δ-Mapping

To control nonphysical oscillations near shocks and discontinuities, a TVB-type minmod slope limiter is applied before each RK stage. For $\delta$8,

$\delta$9

with slope replaced by a δ-mapped minmod,

$$u_t + \partial_x f(u, \theta(x)) = 0, \qquad x \in [0, L],~t > 0,$$0

where $$u_t + \partial_x f(u, \theta(x)) = 0, \qquad x \in [0, L],~t > 0,$$1 maps neighboring cell averages to $$u_t + \partial_x f(u, \theta(x)) = 0, \qquad x \in [0, L],~t > 0,$$2, guaranteeing steady-state preservation.

5. SSP Runge–Kutta Time Integration

Let the semi-discrete DG operator be $$u_t + \partial_x f(u, \theta(x)) = 0, \qquad x \in [0, L],~t > 0,$$3. Time integration is performed by an SSP RK scheme of order $$u_t + \partial_x f(u, \theta(x)) = 0, \qquad x \in [0, L],~t > 0,$$4,

$$u_t + \partial_x f(u, \theta(x)) = 0, \qquad x \in [0, L],~t > 0,$$5

setting $$u_t + \partial_x f(u, \theta(x)) = 0, \qquad x \in [0, L],~t > 0,$$6 at the completion of the step.

The coefficients $$u_t + \partial_x f(u, \theta(x)) = 0, \qquad x \in [0, L],~t > 0,$$7 are taken from the TVD RK (Cockburn–Shu) Butcher tableau.

CFL stability constraint is

$$u_t + \partial_x f(u, \theta(x)) = 0, \qquad x \in [0, L],~t > 0,$$8

with characteristic speeds evaluated on δ-mapped states. The formal accuracy is $$u_t + \partial_x f(u, \theta(x)) = 0, \qquad x \in [0, L],~t > 0,$$9.

6. Algorithmic Workflow

The full algorithm proceeds as:

1. Decompose the domain into cells $u \in \mathbb{R}^r$0 and project $u \in \mathbb{R}^r$1 onto $u \in \mathbb{R}^r$2.
2. At each discontinuous interface, compute δ-mapped states by supply-demand and entropy constraints.
3. Construct the DG spatial operator $u \in \mathbb{R}^r$3 using the unified flux.
4. Apply the δ-mapped slope limiter to suppress oscillations near discontinuities.
5. Advance in time via SSP RK$u \in \mathbb{R}^r$4 under the CFL constraint.

This process achieves high-order accuracy, preserves nontrivial steady states exactly, and suppresses non-physical oscillations arising from flux discontinuities.

7. Representative Applications

The RKDG + δ-mapping methodology has been successfully applied to:

• Nonlinear Elasticity in Layered Media: For systems

$u \in \mathbb{R}^r$5

with periodically varying $u \in \mathbb{R}^r$6, incident pulses split into amplitude-dependent solitary waves; the scheme robustly captures wave speed and structure in agreement with analytic theory.

• Multi-Class Inhomogeneous Traffic Flow: For multiple vehicle classes $u \in \mathbb{R}^r$7, with flux $u \in \mathbb{R}^r$8, the δ-mapping equalizes supply/demand at interface jumps (lane number/speed limit), cleanly resolving up to $u \in \mathbb{R}^r$9 nonlinear waves (shocks, rarefactions, contact discontinuities).

8. Generalization and Extensions

The δ-mapping concept can be integrated into any classical high-order shock-capturing scheme (WENO, TVD finite difference, central schemes), by replacing cell values $\theta(x) \in \mathbb{R}^q$0 by mapped values at discontinuous interfaces. This approach extends naturally to complex network problems, higher-order model couplings, and multidimensional domains with junction or interface conditions, providing a systematic route to high-order, entropy-consistent numerical simulation in the presence of spatially discontinuous coefficients.

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For further technical details, including explicit construction of δ-mapped states, entropy constraints, and specific numerical benchmarks, see (Qiao et al., 2015).