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Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation
L. Pareschi, G. Russo
TL;DR
Stiff relaxation systems require accurate discretizations that work across unresolved stiff and non-stiff regimes without switching solvers. The paper develops high-order IMEX Runge-Kutta schemes with SSP explicit and L-stable DIRK implicit components, and reports AP behavior and accurate computations across regimes.
Problem
Numerical schemes must handle relaxation times ranging from order one to very small values on coarse grids, where solutions approach an equilibrium system and regime-specific solvers are impractical.
Method
The paper derives IMEX Runge-Kutta schemes up to order 3, using SSP explicit discretization, L-stable DIRK treatment of stiff sources, and WENO spatial reconstruction.
Results
The schemes are asymptotic preserving: under det A ≠ 0, the IMEX method becomes the explicit Runge-Kutta scheme for the limit equation, while numerical tests accurately resolve all regimes on coarse meshes.
Takeaways & Limitations
The proposed IMEX-SSP-WENO schemes capture fluid-limit shock formation without spurious oscillations while maintaining prescribed accuracy in stiff and non-stiff limits when no initial layer is present.
Takeaways & Limitations
High-order finite-volume treatment of stiff sources can become impractical because reconstruction couples source terms across cell averages, and reducible implicit tableaux require care for AP claims.
Abstract
from arXiv · showhide
We consider new implicit-explicit (IMEX) Runge-Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strong-stability-preserving (SSP) scheme, and the implicit part is treated by an L-stable diagonally implicit Runge-Kutta methods (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space is obtained by Weighted Essentially Non Oscillatory (WENO) reconstruction. After a description of the mathematical properties of the schemes, several applications will be presented.
1 Introduction
Hyperbolic systems with relaxation span important applications but become challenging when relaxation is stiff and unresolved. The paper develops high-order IMEX Runge-Kutta schemes designed to preserve accuracy and stability across regimes.
- Hyperbolic systems with relaxation model phenomena including rarefied gases, semiconductors, viscoelasticity, multiphase flows, radiation hydrodynamics, traffic flows, and shallow waters.
- Stiffness arises when the relaxation time becomes very small relative to the characteristic time scale, so solutions approach an equilibrium conservation-law system.
- A single solver is needed across stiff and non-stiff regions because separating these regimes is usually extremely difficult or impossible, including on coarse unresolved grids.
- Strang splitting loses its second-order accuracy in the stiff limit for hyperbolic relaxation systems because the relaxation operator has a nontrivial kernel.
- High-order splitting is difficult to use because of stability problems, motivating Runge-Kutta schemes without the same order restriction.
- The paper derives IMEX schemes up to order 3 whose explicit parts are SSP for the limiting conservation-law system and establishes conditions for asymptotic preservation and asymptotic accuracy.
2 IMEX Runge-Kutta schemes
IMEX Runge-Kutta schemes combine explicit treatment of non-stiff terms with implicit treatment of stiff source terms. Their order depends on separate and coupling conditions, while DIRK structure preserves explicit evaluation of the non-stiff component.
- An IMEX Runge-Kutta scheme discretizes source terms implicitly and non-stiff terms explicitly using two coefficient matrices and two weight vectors.
- DIRK treatment of the source terms restricts the implicit matrix to be lower triangular, simplifying the algebraic solves at each stage.
- The double Butcher tableau represents the coupled explicit and implicit Runge-Kutta discretizations, including coefficients for non-autonomous systems.
- Using a DIRK scheme for the source term guarantees that the non-stiff flux is always evaluated explicitly.
- Order conditions are derived by matching one-step Taylor expansions through order Δt^p, with additional coupling conditions arising between the two tableaux.
- For schemes through third order, equal abscissas can simplify the conditions; if the tableaux differ only in A, separate standard conditions ensure third-order accuracy.
- The number of coupling conditions increases dramatically with scheme order, and Table 1 relates these conditions to accuracy.
3 Applications to hyperbolic systems with relaxation
The paper characterizes IMEX Runge–Kutta schemes for hyperbolic relaxation systems, establishing conditions for asymptotic preservation and accuracy in the stiff limit. It shows that, under suitable conditions, the stiff-limit method becomes an explicit Runge–Kutta scheme for the equilibrium system, while SSP structure addresses stability near discontinuities.
- Asymptotic properties: Asymptotic preservation means that, as ε → 0, the IMEX scheme becomes a consistent discretization of the equilibrium equation.Asymptotic accuracy additionally requires retaining the order of accuracy in the stiff limit.
- Asymptotic properties: Nonzero diagonal entries in the DIRK matrix force the relaxation residuals R(U^(i)) to vanish in the stiff limit at every stage.This stagewise equilibrium property underlies the reduction to the limiting conservation-law scheme.
- Limitations and remedies: The stiff-limit accuracy can degrade for non-well-prepared initial data when the first implicit stage does not enforce equilibrium, producing an initial-layer problem.Stiffly accurate implicit schemes may remedy a related final-layer effect for nonconserved quantities.
- Asymptotic properties: If det A ≠ 0 as ε → 0, the IMEX method becomes the explicit Runge–Kutta scheme defined by (˜A, ˜w, ˜c) applied to the limit equation.The equilibrium system is obtained through the conserved variables and the equilibrium map U = E(u).
- Stability and discretization: SSP time discretization and suitable spatial discretization are used to preserve decreasing norms and avoid spurious oscillations near discontinuities.The paper also considers conservative finite-volume and finite-difference spatial discretizations and tabulates several IMEX-SSP schemes.
4 IMEX-WENO schemes
The section develops finite-volume and finite-difference spatial discretizations for IMEX schemes, using reconstruction and numerical fluxes to obtain high-order, non-oscillatory approximations. It also highlights implementation trade-offs for stiff sources and mesh selection.
- Finite volumes: Finite-volume schemes reconstruct cell averages with piecewise polynomials, then evaluate consistent numerical fluxes at cell interfaces.WENO reconstruction is used when discontinuities make high-order reconstruction delicate.
- Finite volumes: Local Lax-Friedrichs flux is cheaper but less accurate than Godunov flux, while higher-order methods reduce the resolution difference.The local Lax-Friedrichs parameter is α = maxw |f′(w)| over the relevant range.
- Finite volumes: High-order finite-volume discretizations can make source terms couple neighboring cell averages, complicating implicit treatment of stiff sources.In relevant relaxation models, conservation properties can sometimes make the implicit step explicitly solvable.
- Finite differences: Finite-difference schemes reconstruct a flux function from pointwise grid values, rather than reconstructing cell averages directly.The flux f is treated as the cell average of a pointwise flux function ˆf before interface reconstruction.
- Finite differences: Finite differences avoid cell coupling because the source is evaluated pointwise, benefiting cases where the implicit step cannot be solved explicitly.Their main mesh limitation is that they require uniform or smoothly varying grids, unlike finite-volume methods on unstructured grids.
- Finite differences: For systems, characteristic-variable reconstruction generally gives better results than conservative-variable reconstruction, which can create small spurious oscillations.The scalar treatment extends to systems by computing the local Lax-Friedrichs parameter from the Jacobian’s spectral radius.
5 Numerical tests
The numerical tests assess convergence, initial-layer behavior, and shock capturing for IMEX-WENO schemes on the Broadwell relaxation model. The schemes retain their prescribed order in nonstiff and stiff limits without initial layers and resolve fluid-limit shocks on coarse meshes.
- Test setup: The tests apply finite-difference WENO schemes with Lax-Friedrichs flux to the Broadwell equations of rarefied gas dynamics.The computations use conservative variables, and less dissipative fluxes could sharpen the numerical results.
- Convergence rates: Without an initial layer, all tested schemes achieve their prescribed accuracy order in both nonstiff and stiff limits, with degradation at intermediate regimes.The comparison is based on relative-error plots for the prototype relaxation system.
- Convergence rates: With an initial layer, ARS(2,2,2) exhibits the expected accuracy degradation.This behavior is consistent with the method’s c1 = 0 treatment of the initial layer.
- Convergence rates: Second-order IMEX-SSP2-WENO and third-order IMEX-SSP3-WENO are evaluated alongside the second-order ARS(2,2,2) method.ARS(2,2,2) is expected to lose accuracy when an initial layer is present because c1 = 0.
- Shock capturing: Both IMEX-SSP-WENO schemes accurately describe the solution across regimes on coarse meshes that do not resolve the small scales.In the fluid limit, shock formation is captured without spurious oscillations.
6 Applications
The applications test IMEX-SSP2-WENO and IMEX-SSP3-WENO on shallow-water, traffic-flow, and granular-gas relaxation problems. Across these examples, the schemes capture stiff-limit behavior and shocks, while higher order generally reduces numerical smearing.
- Application setup: The applications use IMEX-SSP2-WENO and IMEX-SSP3-WENO with 200 grid points, comparing results against reference solutions computed on much finer grids.The reported tests cover several realistic hyperbolic systems with relaxation.
- Shallow water: In the shallow-water model, the zero-relaxation limit is the inviscid Burgers equation, and at ε = 10^-8 the second-order scheme shows pronounced Lax-Friedrichs dissipation.The dissipative effect is less relevant for the third-order scheme.
- Traffic flows: In the traffic-flow test, both schemes produce very similar density profiles at t = 1 for T = 0.2, but the second-order scheme smears the shock more than the third-order scheme.The model combines a vehicle-density continuity equation with a velocity equation involving relaxation toward equilibrium behavior.
- Granular gases: For the granular-gas impact problem, both methods describe the backward-propagating shock, while the second-order method excessively smears the right-boundary layer.The third-order scheme avoids this smearing, although conservative variables produce small spurious pressure oscillations.