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Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit

S. Boscarino, L. Pareschi, G. Russo

arXiv:1110.4375v2math.NAphysics.comp-ph

TL;DR

The paper addresses the parabolic stability restriction that arises when IMEX Runge-Kutta schemes are applied to hyperbolic systems with stiff relaxation in the diffusion limit. It reformulates the problem and applies IMEX schemes so that the limiting diffusion component is treated implicitly. The resulting methodology captures the correct asymptotic behavior, with numerical results showing lower computational cost than approaches yielding explicit diffusion-limit schemes.

  • Problem

    Standard approaches for hyperbolic systems with diffusive relaxation can yield explicit diffusion-limit schemes subject to the parabolic time-step restriction, while robust treatment across rarefied and diffusive regimes remains desirable.

  • Method

    The paper combines the limiting diffusion flux with the convective flux in a reformulated problem and applies IMEX Runge-Kutta schemes to it.

  • Results

    The new approach yields a fully implicit diffusion component in the limiting system, captures the correct asymptotic behavior, and removes the parabolic CFL restriction at lower computational cost than approaches producing explicit diffusion limits.

  • Takeaways & Limitations

    IMEX Runge-Kutta schemes can provide accurate and efficient treatment across rarefied and diffusive regimes while avoiding nonlinear implicit systems through a suitable reformulation and solver blending.

  • Takeaways & Limitations

    Second-order stiffly accurate type A IMEX Runge-Kutta schemes cannot be constructed with three internal stages, requiring additional stages or giving up the FSAL property.

Abstract

from arXiv · show

We consider Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic systems with stiff relaxation in the so-called diffusion limit. In such regime the system relaxes towards a convection-diffusion equation. The first objective of the paper is to show that traditional partitioned IMEX R-K schemes will relax to an explicit scheme for the limit equation with no need of modification of the original system. Of course the explicit scheme obtained in the limit suffers from the classical parabolic stability restriction on the time step. The main goal of the paper is to present an approach, based on IMEX R-K schemes, that in the diffusion limit relaxes to an IMEX R-K scheme for the convection-diffusion equation, in which the diffusion is treated implicitly. This is achieved by an original reformulation of the problem, and subsequent application of IMEX R-K schemes to it. An analysis on such schemes to the reformulated problem shows that the schemes reduce to IMEX R-K schemes for the limit equation, under the same conditions derived for hyperbolic relaxation. Several numerical examples including neutron transport equations confirm the theoretical analysis.

1. Introduction.

Diffusive scaling introduces multiple time scales and a stiff convective contribution, making standard hyperbolic IMEX schemes unsuitable without asymptotic-preserving treatment. The paper proposes reformulating the problem so IMEX Runge-Kutta methods retain a hyperbolic CFL condition while yielding an implicit treatment of diffusion in the limit.

  • Scope: The paper includes applications such as neutron transport and seeks schemes robust across rarefied and diffusive regimes as the scaling parameter varies by orders of magnitude.Numerical tests include comparisons with schemes from the literature.
  • Motivation: Diffusive regimes contain multiple time scales, including a stiff convection term that contributes to the asymptotic diffusion.Schemes must remain consistent and stable for the limit system as ε approaches zero independently of discretization parameters.
  • Motivation: Standard IMEX Runge-Kutta schemes impose Δt = O(εΔx) because the hyperbolic characteristic speed is O(1/ε).In the diffusive regime ε < Δx, this is more restrictive than the parabolic condition Δt = O(Δx^2).
  • Prior approaches: Previous asymptotic-preserving approaches commonly produced explicit schemes for the diffusive limit by splitting the stiff hyperbolic part.The paper instead applies partitioned IMEX schemes with stiffness associated with a variable rather than an operator.
  • Proposed approach: The proposed reformulation combines the limiting diffusion flux with the convective flux, producing IMEX schemes whose diffusion-limit method treats diffusion implicitly.The approach is designed for high-order time accuracy and a broad class of problems.
  • Analysis: The analysis uses power-series expansions in ε and identifies an index-1 differential-algebraic system at leading order.Compatibility between exact and numerical solutions at successive orders imposes additional conditions.

2. Partitioned IMEX R-K schemes.

Traditional partitioned IMEX Runge-Kutta schemes reduce to explicit schemes for the diffusion limit, so they retain the usual parabolic stability restriction. Their construction treats one equation explicitly and the stiff variable equation implicitly, with DIRK structure ensuring explicit evaluation of the nonstiff term.

  • Partitioned formulation: The prototype system has stiffness naturally associated with the variable v, enabling a partitioned Runge-Kutta treatment with the first equation explicit and the second implicit.This is a singular perturbation structure rather than stiffness associated with a single operator.
  • Spatial discretization: The method-of-lines discretization represents the convective and source terms through F(V) = −DV and G(U) = Q(U) − Dp(U).Here D discretizes spatial derivatives, while Q(U) discretizes q(u).
  • Diffusion limit: For L-stable implicit schemes, the IMEX Runge-Kutta method relaxes to the explicit scheme applied to the diffusion-limit equation.The limiting relation is obtained through V_k = G(U_k) and the reduced flux F(G(U_k)).
  • Stability restriction: For p(u) = u and q(u) = 0, the partitioned method becomes explicit Euler for diffusion and obeys the parabolic restriction Δt ⩽ Δx^2/2.Thus the limiting explicit method does not remove the diffusion stability constraint.
  • IMEX structure: The IMEX tableau uses an explicit lower-triangular matrix with zero diagonal and a lower-triangular DIRK matrix for the implicit scheme.This choice guarantees that F(V) is always explicitly evaluated.
  • Scheme classes: Type A schemes require an invertible implicit matrix, whereas CK schemes use an invertible submatrix and include ARS schemes as the special case a = 0.Type CK schemes simplify order conditions, while type A schemes are more amenable to theoretical analysis.

3. Overcoming parabolic stiffness.

The paper reformulates the relaxation system by adding and subtracting a diffusion term, enabling IMEX Runge-Kutta schemes to treat diffusion implicitly in the ε → 0 limit. Under suitable accuracy and stability conditions, the resulting schemes avoid parabolic time-step restrictions while preserving the limit manifold.

  • 3. Overcoming parabolic stiffness.: The reformulation adds and subtracts µp(u)xx, with µ(0) = 1, so diffusion can be treated implicitly while the remaining flux is explicit.The choice µ ≡ 1 gives the largest stability region for the method.
  • 3.1. The new approach.: The BPR approach reduces to an IMEX scheme for the limiting equation, with diffusion implicit and no parabolic stability restriction in the diffusion limit.For q(u) = 0, the limiting method becomes a stiffly accurate DIRK scheme and is unconditionally stable.
  • 3.1. The new approach.: Applying IMEX Runge-Kutta schemes to the reformulated system yields an index-1 differential-algebraic system in the diffusion limit.The analysis assumes an invertible Jacobian gv and represents the constraint through v = G(u).
  • 3.2. Analysis of TYPE A IMEX schemes.: For q(u) ≠ 0, the limiting method retains a nonzero explicit term, treats diffusion implicitly, and requires a classical hyperbolic CFL condition.Stiff accuracy alone projects the numerical solution onto the constraint manifold when the explicit method also satisfies un+1 = Us.
  • 3.2. Analysis of TYPE A IMEX schemes.: Global stiff accuracy makes the numerical solution equal to the last internal stage, ensuring that the ε → 0 solution satisfies g(un+1, vn+1) = 0.The explicit component must additionally be FSAL to guarantee the required endpoint-stage equality in the general case.
  • 3.2. Analysis of TYPE A IMEX schemes.: Type A schemes face construction and order-reduction trade-offs, including the impossibility of a second-order stiffly accurate three-stage scheme.The paper therefore considers a three-stage second-order scheme with b̃ = b, sacrificing FSAL and global stiff accuracy.
  • 3.3. Analysis of TYPE CK schemes.: Type CK schemes similarly yield IMEX discretizations with implicit diffusion and a hyperbolic CFL condition, while globally stiffly accurate schemes project onto the constraint manifold.Their implicit part is an ESDIRK structure with a11 = 0 and common diagonal coefficient for later stages.

4. IMEX-Finite Difference schemes.

The spatial discretization must balance non-oscillatory behavior, dissipation, and stability across hyperbolic and diffusive regimes. Matching discretizations for key derivatives is also required for the correct diffusion limit.

  • Non-oscillatory interpolants such as ENO and WENO are necessary to prevent spurious oscillations near solution discontinuities.
  • Upwind schemes are dissipative because hyperbolic characteristic speeds scale as 1/ε, whereas central differences avoid excessive dissipation but may become unstable.
  • Using the same spatial discretization for ∂xp(u) in both equations, together with µ(0) = 1, preserves the correct diffusion-limit behavior.

1. Correct diffusion limit.

A compact diffusion-limit stencil is possible when the scheme satisfies the stated first condition and uses a suitable discretization of the limiting second derivative.

  • A compact stencil can be obtained in the diffusion limit ε →0 with a suitable discretization of the second-order derivative.The passage identifies this compact-stencil property as an advantage of the approach.

2. Compact stencil.

When q(u) ≠ 0, high-order shock-capturing fluxes are required for convection, including in convection-diffusion limits with small diffusion.

  • Shock-capturing high-order fluxes are needed for the convection part when q(u) ≠ 0 and for all spatial derivatives except the diffusive second-order term.The second-order term µ(ε)∂xxp(u) is treated separately on the right-hand side.

3. Shock capturing.

The implicit derivative ∂xp(u) must use nodal values of u obtained from the first equation, a requirement compatible with WENO-based finite differences.

  • Evaluating the implicit ∂xp(u) derivative from nodal u values supplied by the first equation enables high-accuracy WENO finite-difference discretization.The passage presents WENO reconstruction as an example satisfying the required properties.

4. Avoid solving nonlinear algebraic equations.

The reformulated system separates explicit flux terms from an implicitly treated diffusion contribution, enabling stable spatial discretization across regimes.

  • The scheme treats the reformulated system with an explicit flux and implicitly treated right-hand-side terms.The reformulation adds and subtracts µp(u)xx, allowing the diffusion contribution to be handled implicitly.
  • As ε → 0 and µ → 1, the reformulated system relaxes to the convection-diffusion limit with flux derivative q′(u).The characteristic speed of the limiting equation is λ = q′(u).
  • The numerical method uses conservative finite differences with fluxes reconstructed from positive and negative components.Reconstructions may use high-order ENO or WENO procedures, while derivative terms can be computed pointwise.
  • The local Lax-Friedrichs flux decomposition uses α based on the characteristic speeds of the limiting convection-diffusion equation.The paper cautions that intermediate regimes may have higher characteristic speeds, with stabilization supplied by implicit diffusion.
  • The diffusion term is discretized with second-order finite differences for second-order time schemes and fourth-order finite differences for third-order time schemes.

5. Numerical examples.

Numerical tests assess IMEX Runge-Kutta schemes across convergence, rarefied, intermediate, and diffusive regimes. The results support the expected limiting behavior but reveal accuracy problems for some intermediate-regime discretizations.

  • Scheme selection: The tests use second- and third-order IMEX Runge-Kutta schemes, including the third-order BPR(3,5,3) method introduced in the paper.BPR uses three explicit and five implicit stages and has nonnegative explicit coefficients.
  • Assumptions: All convergence analyses assume well-prepared initial data; otherwise accuracy can decrease unless an initial-layer correction is used.Type A schemes are described as more robust to this issue.
  • Convergence test: The linear diffusion test reaches the expected convergence rates for the u-component.The test uses p(u) = u and q(u) = 0 with the exact solution u(x,t) = u0(x) exp(−t).
  • Convergence test: The convection-diffusion-limit test also reaches the expected convergence rates for the u-component.Table 5.3 reports convergence rates in the L∞-norm for the limit equation.
  • Convergence test: Second-order schemes with central discretization retain their prescribed accuracy uniformly in ε2, whereas WENO32 versions degrade in intermediate regimes.The comparison spans ε2 from 10−6 to 1 with ∆t ≃ ∆x.
  • Convergence test: Third-order WENO53 schemes also lack convergence in intermediate regimes because differently discretized added and subtracted diffusion terms fail to cancel adequately.Matching their spatial discretizations can help in some regimes but may compromise cancellation of q(u)−v−p(u)x in the flux.
  • Shock test cases: The reformulated BPR approach relaxes to an IMEX scheme for the limit equation with explicit convection and implicit diffusion.This is the intended diffusion-limit behavior of the reformulated method.

6. Application to transport equations.

The paper applies the reformulated IMEX Runge-Kutta approach to neutron transport under diffusive scaling, deriving transport discretizations that remain accurate across regimes while treating limiting diffusion implicitly. Numerical tests compare the method with JPT and reference solutions, showing correct diffusion-limit behavior and lower computational cost.

  • Transport model: The section applies the derived IMEX schemes to neutron transport equations under diffusive scaling.The transport model uses mean free path ε and converges to a diffusion equation as ε approaches zero.
  • Reformulation: The even-odd decomposition rewrites transport into equations for the density-like variable r and flux-like variable j, which yield the diffusion equation in the limit.Integrating the first reformulated equation over velocity produces the limiting diffusion equation.
  • Boundary treatment: Boundary conditions are selected to avoid a boundary layer in the limit, while higher-order boundary implementations and other boundary conditions remain outside the study.The numerical boundary treatment uses a second-order implementation of the limiting equation.
  • Reformulation: Adding and subtracting v^2∂xxr/σ produces an equivalent reformulation whose IMEX discretization becomes fully implicit for the limiting diffusion equation.The reformulated system is discretized with IMEX Runge-Kutta integration, combining implicit and explicit contributions.
  • Numerical results: In Problem I, both JPT and BPR closely match the exact diffusive solution, while BPR is about four times faster than the explicit method.The comparison includes transient and steady-state solutions at ε = 10^-8.

7. Conclusions.

The paper concludes that its reformulated IMEX approach handles hyperbolic and kinetic diffusion limits accurately and efficiently. It removes the parabolic CFL restriction for the limiting diffusion component, while extensions beyond IMEX Runge-Kutta methods remain future work.

  • Conclusions: The approach produces accurate and efficient schemes for hyperbolic and kinetic problems in both rarefied and diffusive regimes.The conclusion reports that the schemes capture the correct asymptotic behavior at lower computational cost than approaches yielding explicit diffusion-limit schemes.
  • Conclusions: The reformulation yields a fully implicit method for the diffusion component without solving nonlinear implicit systems.A fully implicit solver for the limiting diffusive system is blended into the IMEX Runge-Kutta method.
  • Conclusions: The method removes the parabolic CFL restriction common to explicit schemes for the underlying diffusion limit.This enables lower computational cost relative to methods with explicit diffusion-limit treatment.
  • Future work: Extensions to more general additive Runge-Kutta schemes and separate implicit solvers for different stiff components are left for future research.The proposed reformulation distinguishes highly stiff equilibrium dynamics from the mildly stiff limiting diffusive term.

8. Appendix.

The appendix analyzes stability and order conditions for IMEX Runge-Kutta schemes. It identifies unconditional diffusive-limit stability for the BPR approach under specified conditions, while proving an impossibility result for a class of second-order type-A schemes.

  • Stability analysis: For the linear case p(u) = u and q(u) = 0, the appendix studies Fourier-mode stability using an IMEX Euler discretization and its stability matrix.The analysis derives eigenvalues and a polynomial condition in the time step.
  • Stability analysis: When ε|ξ| < 1/2, the time-step restriction is parabolic, but the restriction on ∆t/∆x^2 becomes less severe as ε approaches zero.The largest stability region occurs for µ = 1.
  • Stability analysis: Implicit treatment of the second equation stabilizes explicit treatment of the first when ε is sufficiently small.This provides the mechanism for stability in the diffusive regime.
  • Order conditions: Theorem 8.1 states that no second-order two-stage type-A IMEX scheme satisfies the condition b^T A^-1 = e^T \tilde A.The proof derives an impossible γ = 0 condition while requiring A to remain invertible.
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