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Integral Invariants of Vasudeva Murthy's Relaxation Systems: Analysis and Numerical Validation

Sudipta Sahu, Rathan Samala

arXiv:2608.30770v1math.NA

TL;DR

Stiff source terms make relaxation systems difficult to analyze and simulate while preserving stability and transport–relaxation interactions. This paper extends Vasudeva Murthy’s framework to several systems, rigorously establishes integral invariants, and numerically confirms their preservation and robust behavior.

  • Problem

    Stiff source terms introduce multiple time scales, complicating analysis and numerical approximation while requiring simultaneous stability and accurate transport–relaxation resolution.

  • Method

    The study extends Vasudeva Murthy’s alternative relaxation framework to several balance-law systems, derives their limiting formulations, and establishes associated integral invariance properties.

  • Results

    Integral invariance is established for the proposed relaxation models, including the two-dimensional Jin–Xin model, where it provides a conserved quantity supporting physical consistency and numerical robustness.

  • Takeaways & Limitations

    The invariant offers a criterion for evaluating conservation and asymptotic-preserving properties of numerical schemes and connects relaxation models to their limiting conservation laws.

  • Takeaways & Limitations

    The relaxation models require sub-characteristic conditions, such as T0 < γT, to ensure stability and consistency.

Abstract

from arXiv · show

Vasudeva Murthy's relaxation approach [A.S. Vasudeva Murthy, J. Comput. Appl. Math., 203(2), pp. 437-443, 2007], originally proposed for the Jin-Xin relaxation model, provides an alternative formulation with invariant properties that is consistent and retains the semilinear structure incomparison to the standard one. In this work, Vasudeva Murthy's relaxation approach for various relaxation systems are proposed such as the shallow water equations, the Broadwell model, the Euler equations with heat transfer and two-dimensional Jin-Xin model. For proposed relaxation models, the associated integral invariants are rigorously established at the theoretical level. The main advantage of the integral invariant is that it provides a conserved quantity for the relaxation system by incorporating the coupled contributions of the solution variables in vector form. To validate the analytical results, numerical simulations are carried out for each model using three second-order numerical schemes: CS-EBT2, a semi-implicit second-order central finite-volume scheme for hyperbolic systems with relaxation source terms [S. Sahu, E. Macca, and R. Samala, J. Comput. Phys., 563, 115100, 2026]; UCS2, a finite-volume central relaxation-type scheme [S. F. Liotta, V. Romano, and G. Russo, SIAM J. Numer. Anal., 38(4), 1337-1356, 2000]; and IMEX-RK2, a second-order Implicit-Explicit Runge-Kutta scheme [Pareschi and Russo, J. Sci. Comput., 25, 129-155, 2005]. Numerical results, compared to exact or finely resolved reference solutions, confirm that the models preserve integral invariants, remain stable under CFL restrictions, and exhibit robust and accurate behavior across all benchmark systems tested.

1. Introduction

The introduction frames stiff hyperbolic balance laws as challenging multiscale problems requiring stable, accurate treatment of transport–relaxation interactions. It motivates extending Vasudeva Murthy’s alternative relaxation framework to several one- and two-dimensional systems.

  • Problem setting: Stiff source terms create multiple time scales, making mathematical analysis and numerical approximation difficult while requiring simultaneous stability and accurate transport–relaxation coupling.These systems model phenomena combining convective transport and relaxation mechanisms in areas including compressible fluid dynamics, kinetic theory, and shallow-water flows.
  • Numerical background: The introduction places the work within finite-volume central schemes and IMEX Runge–Kutta methods, which offer robust spatial discretization or stable treatment of stiff source terms.Central schemes avoid Riemann solvers and characteristic decomposition, whereas IMEX methods treat fluxes explicitly and stiff sources implicitly.
  • Motivation and contribution: The study extends Vasudeva Murthy’s alternative relaxation framework from the one-dimensional linear Jin–Xin model to shallow-water, Broadwell, Euler-with-heat-transfer, and two-dimensional Jin–Xin systems.The paper develops alternative relaxation models for these representative hyperbolic balance laws and establishes their integral invariance properties.
  • Scope: Although the primary focus is one-dimensional problems, the framework is generalized to multidimensional balance laws relevant to shallow-water flows and gas dynamics.The two-dimensional formulation uses separate fluxes in the x- and y-directions.

2. The Alternative Relaxation Systems and Their Asymptotic Limits

The section defines relaxation systems through conserved variables and equilibrium dynamics, then analyzes Vasudeva Murthy’s alternative Jin–Xin formulation and extends its structure to shallow water, Broadwell, and Euler systems with heat transfer. Chapman–Enskog limits, sub-characteristic conditions, and exact integral invariants establish asymptotic consistency, stability, and conservation across these models.

  • General relaxation framework: Relaxation systems require a rank-deficient matrix Q satisfying QR(U)=0, yielding conserved variables v=QU and an equilibrium system with flux G(v)=QF(E(v)).When τ is small, stiff relaxation drives solutions rapidly toward the equilibrium manifold.
  • Alternative Jin–Xin model: The alternative Jin–Xin model [14] has a Chapman–Enskog diffusive limit governed by η(u)=1−a^2, with η(u)≥0 ensuring parabolicity, stability, and consistency.Its exact conservation law defines an integral invariant that couples u_τ and v_τ, remains valid for all t, and reduces to the limiting conservation-law invariant as τ→0.
  • Shallow Water Equations: The alternative shallow-water relaxation model has a parabolic diffusive limit under 0<h_τ<4 and preserves a conservative identity that yields an integral invariant.The relaxation term enforces the equilibrium relation h_τu_τ=1/2 h_τ^2 as τ→0.
  • Broadwell model: The proposed Broadwell relaxation model retains a conservative structure, while its diffusive correction stabilizes the system under Liu’s sub-characteristic condition.The cited passages establish parabolic behavior and stabilization but do not state the condition’s explicit formula.
  • Euler equations with heat transfer: The Euler relaxation model with heat transfer converges as τ→0 to compressible Euler equations with an equilibrium temperature constraint and preserves mass, momentum, and an exact integral invariant.The model is stable and consistent when T_0<γT, and its parabolic diffusive limit describes thermal relaxation.

Relaxation Approximation for Two-Dimensional Hyperbolic Balance Laws:

The proposed alternative two-dimensional Jin–Xin relaxation model extends the framework to hyperbolic balance laws, remains formally consistent with the underlying conservation law, and preserves an integral invariant under suitable boundary conditions.

  • Relaxation Approximation for Two-Dimensional Hyperbolic Balance Laws:: The framework extends the proposed relaxation approach to two-dimensional hyperbolic balance laws with conserved variables, directional fluxes, and source terms.The formulation introduces relaxation variables associated with the x- and y-direction fluxes.
  • Relaxation Approximation for Two-Dimensional Hyperbolic Balance Laws:: As τ → 0, stiff source terms drive auxiliary variables toward equilibrium, recovering the underlying two-dimensional hyperbolic conservation law.The reduced conservation law obtained from the equilibrium relations establishes formal asymptotic consistency.
  • Relaxation Approximation for Two-Dimensional Hyperbolic Balance Laws:: The τ-expansion yields a parabolic-type reduced equation whose diffusion coefficients are positive when the Liu sub-characteristic conditions hold.These conditions ensure stability of the relaxation approximation and consistency with the underlying hyperbolic system.
  • Relaxation Approximation for Two-Dimensional Hyperbolic Balance Laws:: The alternative two-dimensional Jin–Xin relaxation model satisfies a conservation law and preserves the corresponding integral invariant under periodic or homogeneous zero-flux boundary conditions.The invariant supplies a physically meaningful conserved quantity that supports the model’s physical consistency and numerical robustness.

3. Numerical Approximation and Validation · 3.1. Second Order FV Central Schemes.

The numerical validation uses three second-order schemes—CS-EBT2, UCS2, and IMEX RK2 —to test one- and two-dimensional relaxation systems for accuracy, robustness, diffusive-limit consistency, and integral-invariant preservation. The finite-volume central schemes use staggered-grid reconstructions with implicit or semi-implicit treatment of stiff relaxation sources, including a newly reported two-dimensional UCS2 formulation.

  • 3. Numerical Approximation and Validation: Three second-order schemes, CS-EBT2, UCS2, and IMEX RK2, are applied to one- and two-dimensional benchmarks assessing accuracy, robustness, diffusive-limit consistency, and integral-invariant preservation.The methods validate the alternative relaxation formulations developed previously.
  • 3.1. Second Order FV Central Schemes.: CS-EBT2 and UCS2 discretize hyperbolic balance laws with staggered-grid finite volumes, Riemann-solver-free central fluxes, and semi-implicit relaxation sources to achieve second-order accuracy in space and time.The framework is based on the Nessyahu–Tadmor staggered-grid construction.
  • 3.1. Second Order FV Central Schemes.: Piecewise linear reconstruction with a MinMod limiter provides non-oscillatory updates while retaining second-order accuracy in smooth regions.Flux contributions use staggered half-time predictor states and midpoint temporal quadrature.
  • 3.1.1. One Dimensional CS-EBT2 Scheme.: The one-dimensional CS-EBT2 predictor-corrector scheme treats stiff sources implicitly, preserves the conservative structure and integral invariants under suitable boundary conditions, and has a larger stability region permitting larger CFL numbers.Its source integral is approximated using a semi-implicit backward Taylor expansion.
  • 3.1.2. One Dimensional UCS2 Scheme.: The one-dimensional UCS2 predictor-corrector scheme combines Radau source quadrature with implicit treatment of the stiff contribution, preserves conservation and integral invariants, but generally requires more restrictive CFL conditions than CS-EBT2.Predictor states supply the first source evaluations, while the final term is treated implicitly at the new time level.
  • 3.1.3. Two Dimensional CS-EBT2 Scheme.: The two-dimensional extension uses Cartesian-cell averages, MinMod-limited reconstruction in both directions, four-face numerical fluxes, and corresponding semi-implicit or Radau-based source updates.The construction extends the one-dimensional finite-volume formulation to x- and y-direction fluxes.
  • 3.1.4. Two Dimensional UCS2 Scheme.: The two-dimensional UCS2 formulation extends the UCS2 scheme proposed in with the same finite-volume construction and Radau source treatment, reportedly for the first time in the literature.The update uses staggered averages and MinMod-limited x- and y-direction slopes.

3.2. Second-Order IMEX Runge-Kutta Scheme.

The section formulates a second-order IMEX Runge–Kutta method for relaxation balance laws, treating convection explicitly and relaxation sources implicitly. Its stiff accuracy supports stability for vanishing relaxation parameters and consistency with equilibrium limits in one and two dimensions.

  • One-dimensional formulation: The IMEX RK2 scheme treats convective terms explicitly and relaxation source terms implicitly at Runge–Kutta stages, addressing their different time scales.The method follows Pareschi and Russo and the Boscarino reformulation.
  • One-dimensional formulation: The one-dimensional discretization uses conservative finite volumes with second-order piecewise-linear MinMod reconstruction before IMEX time integration.This combines spatial slope limiting with the IMEX treatment of the balance-law source term.
  • One-dimensional formulation: Because the final implicit stage equals the numerical solution, the scheme is stiffly accurate, stable as relaxation parameters vanish, equilibrium-consistent, and conservative.These properties are stated for the one-dimensional formulation and follow from the final-stage construction.
  • Two-dimensional formulation: The two-dimensional extension uses Cartesian finite volumes with fluxes in both coordinate directions, applying the same second-order MinMod reconstruction and IMEX RK2 coefficients.Only the spatial operator changes from one to two dimensions; convection remains explicit and relaxation remains implicit.
  • Two-dimensional formulation: In the two-dimensional scheme, implicit source updates are local to each cell, so spatial coupling occurs only through numerical fluxes.The spatial operator accounts for fluxes through the four faces of each Cartesian control volume.

3.3. Numerical Experiments.

Numerical experiments evaluate the proposed relaxation models with CS-EBT2, UCS2, and IMEX RK2 against exact or high-resolution reference solutions. Across benchmark problems, figures and L1-error studies assess solution behavior, convergence, and preservation of integral invariance.

  • 3.3. Numerical Experiments.: The three schemes are evaluated on uniform grids against exact solutions when available or high-resolution IMEX RK2 reference solutions otherwise.The IMEX RK2 reference uses a finer mesh, and the time step is selected to ensure stability through CFL restrictions.
  • 3.3. Numerical Experiments.: Integral invariance is assessed for all conserved variables using the discrete L1-norm and grid-refinement studies.Minimal variation in the L1-norm across successive resolutions indicates that the numerical schemes maintain the invariant property.
  • 3.3. Numerical Experiments.: Figures and tables demonstrate that the discretized relaxation models retain the expected conservation properties across the benchmark problems.Figures show solution evolution under different relaxation parameters, while tables report corresponding L1-errors for integral-invariance preservation.
  • 3.3. Numerical Experiments.: Two-dimensional benchmarks use uniform Cartesian meshes, directional convective fluxes, scheme-specific relaxation-source discretizations, and a multidimensional CFL condition.Solutions are compared with exact solutions when available or reference solutions otherwise.
  • 3.3. Numerical Experiments.: For the smooth alternative Jin–Xin benchmark, Tables 1 and 2 report L1-errors and convergence rates for u and v under τ = 10−10 and CFL = 0.9.The computations use N = 320 points, a = 0.7, periodic boundary conditions, and final time T = 0.35; Figure 1 compares all schemes with the exact solution.

N CS-EBT2 UCS2 IMEX RK2

Across alternative Jin–Xin and modified shallow-water relaxation models, CS-EBT2, UCS2, and IMEX-RK2 preserve the integral invariant and produce accurate numerical solutions under stiff-relaxation tests. Simulations compare numerical results with exact or high-resolution reference solutions across specified grids, CFL conditions, boundary conditions, and final times.

  • Numerical setup: The Jin–Xin simulations use N = 400 uniformly spaced points on [0, 1] with transmissive boundary conditions and compare each scheme against the corresponding exact solution.The shallow-water tests use [−1, 1] with CFL 0.9 and periodic boundary conditions, while another stiff test uses N = 200, transmissive boundaries, and T = 0.5 against a high-resolution reference solution.
  • Alternative Jin–Xin model: All three schemes preserve the integral invariant for the alternative Jin–Xin model across relaxation parameters τ = 10^-10, 0.02, and 1.Figures 2–5 report comparisons at T = 0.35, while Table 3 verifies the invariant for u + τv at τ = 10^-10 and CFL 0.9.
  • Modified shallow-water model: CS-EBT2, UCS2, and IMEX-RK2 demonstrate accuracy for the modified shallow-water relaxation model in the stiff regime τ = 10^-10.The experiments use CFL 0.9, periodic boundaries, and final time T = 0.35; Tables 4 and 5 report L1-errors and corresponding convergence information.

N CS-EBT2 UCS2 IMEX RK2

The section reports error and convergence-order evaluation for hu in the modified shallow water model under a smooth initial condition with τ = 10−10.

  • Table 5 evaluates error and order of convergence for hu in the modified shallow water model with smooth initial condition (3.40) and τ = 10−10.

N CS-EBT2 UCS2 IMEX RK2

The CS-EBT2, UCS2, and IMEX RK2 schemes preserve the integral invariant of the alternative shallow water model. The section also evaluates these schemes on alternative Broadwell data using error and convergence measurements.

  • Alternative shallow water model: CS-EBT2, UCS2, and IMEX RK2 preserve the integral invariant h + τhu for the alternative shallow water model.The test uses τ = 10−10 and CFL 0.9.
  • Alternative shallow water model: The alternative shallow water invariant test is reported for h + τhu using the three schemes’ errors and computed invariant values.Table 6 specifies the initial condition, τ = 10−10, and CFL 0.9.
  • Alternative Broadwell model: For the alternative Broadwell model, simulations use conservative variables ρ, m, and z with periodic boundaries for smooth data and transmissive boundaries for non-smooth data.All schemes use CFL 0.9, τ = 10−8, and final time T = 0.5.
  • Alternative Broadwell model: The Broadwell smooth-case experiments report L1-error norms and corresponding convergence orders for ρ, m, and z.Tables 7, 8, and 9 present the measurements for the three conservative variables.

N CS-EBT2 UCS2 IMEX RK2 · N CS-EBT2 UCS2 IMEX RK2

The numerical tests assess CS-EBT2, UCS2, and IMEX RK2 on alternative relaxation models, including Broadwell, Euler equations with heat transfer, and the two-dimensional Jin–Xin system. Across the reported tests, the schemes preserve integral invariants under specified CFL and relaxation-parameter settings, while the 2D Jin–Xin smooth case achieves near-machine-precision invariant errors.

  • N CS-EBT2 UCS2 IMEX RK2: The alternative Broadwell tests include an integral-invariance evaluation for ρ and m + τz at τ = 10^-8 and CFL 0.9.The associated error and convergence data for the modified shallow water model are also reported for z with τ = 10^-8.
  • N CS-EBT2 UCS2 IMEX RK2: For the alternative Broadwell model, the schemes use transmissive boundaries, CFL 0.9, and final time T = 0.5 while testing τ = 10^-8, 0.02, and 1 against an IMEX RK2 reference.The results include comparisons of ρ, m, and z for the non-smooth case, with τ = 10^-8 shown in Figure 7 and larger relaxation parameters in Figures 8 and 9.
  • 3.3.4. Euler Equations with Heat Transfer.: For the alternative Euler equations with heat transfer, CS-EBT2, UCS2, and IMEX RK2 are compared on N = 200 points with a finer-grid IMEX RK2 reference solution.The reported non-smooth test uses transmissive boundaries and compares ρ, ρu, and ρE at CFL 0.9 with τ = 10^-8.
  • 3.3.4. Euler Equations with Heat Transfer.: The alternative Euler heat-transfer model evaluates the conserved variables density, velocity, and total energy through the prescribed relaxation formulation.The reported numerical variables are ρ, ρu, and ρE, and the tests use the CS-EBT2, UCS2, and IMEX RK2 schemes.
  • 3.3.5. 2D Jin-Xin relaxation model.: The 2D Jin–Xin smooth case compares exact and numerical contour plots for u, v, and w on a 400 × 400 grid at CFL 0.9, τ = 10^-10, and T = 0.35.The exact solutions are obtained analytically using the Fourier transform method, and Tables 12–14 report observed convergence orders for the three variables.
  • 3.3.5. 2D Jin-Xin relaxation model.: For the 2D Jin–Xin smooth test, all three schemes preserve the discrete integral invariant across mesh resolutions with L1-errors nearly unchanged and close to machine precision.The computations use CFL 0.9, τ = 10^-10, a = 0.7, and b = 0.5; the result supports the theoretical conservation property.
  • 3.3.5. 2D Jin-Xin relaxation model.: The 2D Jin–Xin study also reports integral-invariance data for u + τv + τw at τ = 10^-10 and CFL 0.9.The smooth well-prepared initial condition lies on the equilibrium manifold, producing smooth evolution suitable for assessing accuracy and stability.

4. conclusion

The study establishes and numerically verifies integral-invariant preservation for Vasudeva Murthy’s relaxation formulations across one-dimensional benchmark systems and a proposed two-dimensional Jin–Xin extension. Second-order schemes effectively apply to these models and support the invariant-preserving conclusions.

  • 4. conclusion: The relaxation approach is examined for shallow-water equations, the Broadwell model, and Euler equations with heat transfer in one dimension, alongside a two-dimensional Jin–Xin extension.The approach was originally introduced for the Jin–Xin relaxation model.
  • 4. conclusion: Computational verification uses the second-order CS-EBT2, UCS2, and IMEX-RK2 schemes, with CS-EBT2 treating source terms implicitly and UCS2 using predictor-stage evaluations.CS-EBT2 and UCS2 extend the NT central scheme to non-homogeneous balance laws.
  • 4. conclusion: Numerical experiments confirm preservation of the derived integral invariants across one-dimensional relaxation models and the two-dimensional Jin–Xin model.The results provide theoretical and numerical evidence for the invariant-preserving character of the considered formulations.
  • 4. conclusion: The established numerical schemes are effectively applicable to the benchmark models, including the proposed two-dimensional extension.

Funding

Both authors received support from the NBHM, DAE, India project under Ref. No. 02011/46/2021 NBHM(R.P.)/R&D II/14874.

  • Funding: Both authors were supported by the NBHM, DAE, India project, Ref. No. 02011/46/2021 NBHM(R.P.)/R&D II/14874.
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