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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 the mathematical analysis and numerical approximation of relaxation systems more challenging. This paper extends an alternative relaxation framework to several systems and establishes integral invariants for the proposed models.

  • Problem

    Stiff source terms introduce multiple time scales, complicating analysis and approximation while requiring numerical methods to maintain stability and accurately capture transport-relaxation interactions.

    Source: §1 · p. 1

  • Method

    The paper extends the alternative relaxation framework to several hyperbolic balance law systems and establishes the associated integral invariance properties for the proposed models.

    Source: §1 · p. 2, §2.1 · p. 4

  • Results

    Integral invariance is established for the proposed relaxation models, including the alternative Jin–Xin and two-dimensional Jin–Xin models.

    Source: §1 · p. 2, §2.1 · p. 4, §2.5 · p. 8

  • Takeaways & Limitations

    The integral invariant provides a conserved quantity that accounts for coupled solution-variable contributions and serves as a criterion for assessing numerical conservation.

    Source: §2.1 · p. 4

    The Euler heat-transfer relaxation model's stability and consistency require the sub-characteristic condition T0 < γT.

    Source: §2.4 · p. 7

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

This study extends Vasudeva Murthy’s alternative relaxation framework to several one- and two-dimensional systems, proves integral invariance for the proposed models, and assesses them numerically.

Introduction

  • Stiff source terms introduce multiple time scales, making analysis and approximation challenging and requiring numerical methods to balance stability with accurate transport-relaxation coupling.Hyperbolic balance laws with stiff sources model phenomena across compressible fluid dynamics, kinetic theory, and shallow-water flows.Source: §1 · p. 1
  • Central schemes CS-EBT2 and UCS2 are Riemann-solver-free finite-volume methods, while IMEX Runge-Kutta methods treat fluxes explicitly and stiff sources implicitly.The central schemes build on Nessyahu-Tadmor and retain simplicity while aiming for accurate, essentially non-oscillatory solutions.Source: §1 · pp. 1–2
  • Vasudeva Murthy’s 2007 alternative for linear Jin-Xin preserves semilinear structure and an integral invariant, alongside a sub-characteristic condition, an L∞ estimate, and convex entropy.The vector-form invariant combines coupled solution-variable contributions and connects the relaxation system with its limiting model as the relaxation parameter tends to zero.Source: §1 · p. 2
  • The study develops alternative relaxation formulations for shallow-water, Broadwell, and Euler-with-heat-transfer systems, extends the methodology to two-dimensional Jin-Xin, and rigorously establishes each model’s integral invariance.The one-dimensional linear Jin-Xin formulation was established previously in.Source: §1 · p. 2
  • Numerical experiments use CS-EBT2, UCS2, and IMEX Runge-Kutta, comparing with analytical solutions where available and highly resolved IMEX references otherwise; results indicate accurate, robust approximations.The experiments assess both one- and two-dimensional hyperbolic balance law systems.Source: §1 · p. 2
  • Section 2 presents the framework and invariant proofs, Section 3 describes methods and computational experiments, and Section 4 gives concluding remarks.Source: §1 · p. 2

2. The Alternative Relaxation Systems and Their Asymptotic Limits

The section defines relaxation systems through conserved variables v = QU and an equilibrium mapping U = E(v), which yields a reduced system and rapid convergence toward equilibrium when τ is small. It applies this framework to Jin–Xin, shallow-water, Broadwell, and Euler systems, deriving diffusive limits and integral invariants for the proposed alternatives.

2.1. Jin-Xin Relaxation Approximation:

  • The alternative Jin-Xin relaxation model has an exact conservation law: its integral invariant accounts for coupled contributions of uτ and vτ and reduces to the corresponding integral invariant of the limiting conservation law as τ → 0.Source: §2.1 · p. 4

2.2. Shallow Water Equations:

  • The equation is parabolic under Liu’s sub-characteristic condition 0 < hτ < 4, and the relaxation system’s conservative identity shows that dx is preserved in time, making it an integral invariant.Source: §2.2 · p. 5

2.3. Broadwell System:

  • For the Broadwell system [21, 13], the diffusive limit is parabolic under Liu’s sub-characteristic condition, and its conservative structure implies an integral invariant.The relaxation approximation’s sub-characteristic condition also ensures hyperbolicity and stability.Source: §2.3 · pp. 5–6

2.4. Euler Equations with Heat Transfer:

  • For Euler equations with heat transfer [22], the diffusive limit is parabolic when T0 < γT, and the alternative model’s conservative structure implies an integral invariant.In the stiff-relaxation limit, the system reduces to the compressible Euler equations with an equilibrium temperature constraint.Source: §2.4 · pp. 6–7

Relaxation Approximation for Two-Dimensional Hyperbolic Balance Laws:

The section extends the relaxation framework to two-dimensional hyperbolic balance laws and presents a two-dimensional Jin–Xin model whose relaxation limit is consistent with the underlying law and whose integral invariant is preserved under suitable boundary conditions.

Relaxation Approximation for Two-Dimensional Hyperbolic Balance Laws:

  • The framework is extended to two-dimensional hyperbolic balance laws with conserved variables, directional fluxes, and a source term, motivating a corresponding two-dimensional relaxation system.The flux functions represent the x- and y-directions, respectively.Source: Relaxation Approximation for… · p. 7

2.5. 2D Jin-Xin relaxation model:

  • As τ tends to zero, the two-dimensional Jin–Xin relaxation system recovers the underlying hyperbolic conservation law through its equilibrium relations.The passage characterizes this asymptotic limit as formal consistency with the original law.Source: §2.5 · pp. 7–8
  • An alternative relaxation approximation is introduced for the two-dimensional Jin–Xin model.The model is supplemented with initial conditions for the conserved and auxiliary relaxation variables.Source: §2.5 · p. 8
  • The leading-order reduced equation is parabolic under the Liu sub-characteristic conditions, which ensure positive diffusion coefficients and support stability and consistency.The reduced equation includes diffusion contributions in both spatial directions.Source: §2.5 · p. 8
  • The alternative two-dimensional Jin–Xin relaxation model preserves its corresponding integral invariant when periodic or homogeneous zero-flux boundaries make boundary integrals vanish.The invariant combines the model’s solution variables in an integral conservation law.Source: §2.5 · p. 8

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

The section presents second-order schemes for validating the proposed relaxation formulations on one- and two-dimensional benchmark problems. The tests assess accuracy, robustness, consistency with the diffusive limit, and preservation of integral invariants.

3. Numerical Approximation and Validation

  • CS-EBT2, UCS2, and IMEX RK2 are used to evaluate one- and two-dimensional relaxation systems across benchmark problems.The stated assessment covers accuracy, robustness, consistency with the diffusive limit, and preservation of associated integral invariants.Source: §3 · p. 9

3.1. Second Order FV Central Schemes.

  • CS-EBT2 and UCS2 use staggered-grid finite volumes with Riemann-solver-free central fluxes and semi-implicit source treatment for second-order accuracy in space and time.The framework follows Nessyahu and Tadmor and is intended to provide stable, robust approximations with stiff source terms.Source: §3.1 · p. 9

3.1.1. One Dimensional CS-EBT2 Scheme.

  • CS-EBT2 uses an implicit source treatment in its predictor-corrector scheme and preserves the conservative structure and, under suitable boundary conditions, integral invariants.It is described as second-order in space and time; reports a larger stability region, lower CPU time, and simpler implementation.Source: §3.1.1 · p. 10

3.1.2. One Dimensional UCS2 Scheme.

  • UCS2 uses Radau quadrature with predictor evaluations and an implicit new-time-level term, preserving conservation and integral invariants under appropriate boundary conditions.It is second-order in space and time, but has a smaller stability region and higher computational cost than CS-EBT2 [10].Source: §3.1.2 · p. 10

3.1.3. Two Dimensional CS-EBT2 Scheme.

  • The two-dimensional CS-EBT2 formulation applies semi-implicit source discretization to the finite-volume update, with fluxes across four cell faces and a space-time source approximation.The construction uses piecewise-linear reconstruction and MinMod-limited slopes in both spatial directions.Source: §3.1.3 · p. 11, §3.1.2 · p. 11
  • The paper extends UCS2 to an explicit two-dimensional formulation using Radau quadrature for the source contribution, reporting it as the first such formulation in the literature to the authors’ knowledge.Source: §3.1.4 · p. 11

3.1.4. Two Dimensional UCS2 Scheme.

  • The two-dimensional UCS2 update uses staggered averages in both directions and MinMod-limited slopes for its spatial reconstruction.Its two-dimensional source treatment follows the Radau quadrature formulation described for UCS2.Source: §3.1.4 · pp. 11–12

3.2. Second-Order IMEX Runge-Kutta Scheme.

The second-order IMEX RK2 scheme treats convective terms explicitly and relaxation sources implicitly, applying the same time discretization to one- and two-dimensional balance laws. Its formulations combine finite-volume spatial operators with a stiffly accurate time update suitable for stiff relaxation.

3.2. Second-Order IMEX Runge-Kutta Scheme.

  • IMEX RK2 treats convection explicitly and relaxation sources implicitly, easing severe time-step restrictions from stiff sources while providing suitable stability in the stiff-relaxation regime.The same second-order time discretization is used in one and two dimensions; spatial operators vary with the problem dimension, but Runge–Kutta coefficients do not.Source: §3.2 · p. 12

3.2.1. One dimensional IMEX Runge-Kutta Scheme.

  • The scheme uses a conservative finite-volume formulation with second-order piecewise-linear MinMod reconstruction, and a Boscarino reformulation introduces an auxiliary explicit state to evaluate the second-stage explicit flux.The numerical solution at time tn+1 coincides with the final implicit stage, making the scheme stiffly accurate; it remains stable as relaxation parameters vanish, is consistent with the equilibrium limit, and preserves the balance law’s conservative structure.Source: §3.2.1 · pp. 12–13

3.2.2. Two-Dimensional IMEX Runge–Kutta Scheme.

  • The two-dimensional formulation uses the same IMEX RK2 coefficients and MinMod reconstruction in both directions, with fluxes through each cell’s four faces defining its spatial operator.Convection remains explicit and relaxation sources implicit; the final-stage solution retains stiff accuracy, and source updates are solved locally in each cell.Source: §3.2.2 · p. 13

3.3. Numerical Experiments.

Numerical experiments evaluate the proposed relaxation models with three schemes and report that they retain expected conservation properties across the tested relaxation regimes and initial conditions. The section also describes how conservation is assessed and presents a modified Jin–Xin benchmark.

3.3. Numerical Experiments.

  • CS-EBT2, UCS2, and IMEX RK2 solutions are compared with exact solutions when available or with finer-mesh IMEX RK2 references.CFL numbers are chosen to ensure numerical stability for each scheme.Source: §3.3 · p. 14
  • Integral invariance is assessed for all conserved variables using the discrete L1-norm, with grid refinement tracking its variation as resolution increases.Minimal L1-norm variation across successive resolutions indicates that the numerical schemes effectively maintain the property.Source: §3.3 · p. 14
  • The benchmark results indicate that the discretized proposed models retain expected conservation properties across the considered relaxation regimes and initial conditions.Figures show solution-variable evolution under different relaxation parameters, while tables report corresponding L1 errors.Source: §3.3 · p. 14

3.3.1. Jin-Xin Relaxation Model.

  • For the alternative Jin–Xin model with smooth (well-prepared) initial data, computations use a = 0.7, N = 320, CFL = 0.9, periodic boundaries, τ = 10−10, and final time T = 0.35.Figure 1 compares CS-EBT2, UCS2, and IMEX RK2 solutions with the corresponding exact solutions for u and v.Source: §3.3.1 · p. 14, Figure 1 · p. 14
  • For smooth modified Jin–Xin data, Tables 1 and 2 report scheme-wise L1 errors and convergence rates for u and v at τ = 10^-10 and CFL = 0.9.The simulations use initial data (3.38).Source: §3.3.1 · p. 14, Table 1 · p. 15, Table 2 · p. 15

N CS-EBT2 UCS2 IMEX RK2

The numerical tests report preservation of the integral invariant by CS-EBT2, UCS2, and IMEX RK2, and assess their accuracy for the modified shallow-water relaxation model.

3.3.2. Shallow Water Equations.

  • For the modified shallow-water relaxation model, the stiff-regime tests report L1 errors and convergence orders for h and hu using all three schemes.The smooth initial-condition tests use τ = 10^-10, CFL 0.9, periodic boundaries, and final time T = 0.35.Source: §3.3.2 · p. 15, Table 4 · p. 17

N CS-EBT2 UCS2 IMEX RK2

  • For the alternative Jin–Xin model with initial condition (3.39), Table 3 verifies that CS-EBT2, UCS2, and IMEX RK2 preserve the integral invariant for u + τv at τ = 10^-10 and CFL 0.9.Source: N CS-EBT2 UCS2 IMEX RK2 · p. 15, Table 3 · p. 16

3.3.2. Shallow Water Equations.

  • In stiff-regime shallow-water tests, the three schemes’ h and hu solutions on 200 grid points are compared with an IMEX RK2 reference on 3200 points, while Table 6 verifies integral invariance.The simulations use transmissive boundaries, τ = 10^-10, CFL 0.9, and final time T = 0.5.Source: §3.3.2 · p. 16

N CS-EBT2 UCS2 IMEX RK2

The section reports an error and convergence-order assessment for the modified shallow-water model with a smooth initial condition.

N CS-EBT2 UCS2 IMEX RK2

  • Table 5 reports error and order of convergence for hu with τ = 10^-10 under smooth initial condition (3.40) and τ = 10^-10.The caption does not provide the table’s numerical values.Source: Table 5 · p. 17

N CS-EBT2 UCS2 IMEX RK2

The reported schemes preserve the alternative shallow-water model’s integral invariant, and attain the expected second-order accuracy for the alternative Broadwell model. The results are reported under specified relaxation and CFL conditions.

N CS-EBT2 UCS2 IMEX RK2

  • CS-EBT2, UCS2, and IMEX RK2 preserve the alternative shallow-water model’s integral invariant for h + τhu at τ = 10−10 and CFL 0.9.Table 6 concerns initial condition (3.41).Source: N CS-EBT2 UCS2 IMEX RK2 · p. 17, Table 6 · p. 18
  • For the NT scheme, τ → 0 permits larger CFL numbers without loss of stability, whereas larger τ requires a more restrictive CFL condition for stability and accuracy.Source: N CS-EBT2 UCS2 IMEX RK2 · p. 18

3.3.3. Broadwell System.

  • The alternative Broadwell initial data use conservative variables ρ, m, and z, with periodic boundaries for smooth data and transmissive boundaries for non-smooth data.The non-smooth case introduces a discontinuity at x = 0.2.Source: §3.3.3 · p. 18
  • CS-EBT2, UCS2, and IMEX RK2 attain the expected second-order accuracy for the alternative Broadwell model.The smooth-case computation uses periodic boundaries on [−1, 1], final time T = 0.5, CFL 0.9, and τ = 10−8.Source: §3.3.3 · p. 18

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

Numerical tests assess integral invariance across the relaxation models and compare computed solutions with reference or exact solutions. The 2D Jin–Xin tests also report convergence results for u, v, and w.

N CS-EBT2 UCS2 IMEX RK2

  • Table 9 presents error and convergence-order results for z in the modified shallow water model with smooth initial condition (3.42) and τ = 10^-8.Source: Table 9 · p. 19
  • For the alternative Broadwell model, numerical solutions are compared with an IMEX RK2 reference at τ = 10^-8, 0.02, and 1; Table 10 reports an integral-invariance examination.The tests use transmissive boundaries, domain [−1, 1], CFL 0.9, and final time T = 0.5.Source: Table 10 · p. 19, N CS-EBT2 UCS2 IMEX RK2 · p. 20, Figure 7 · p. 20

3.3.4. Euler Equations with Heat Transfer.

  • For the alternative Euler equations with heat transfer, CS-EBT2, UCS2, and IMEX RK2 preserve the integral invariant in the reported numerical tests.Solutions on N = 200 are compared with an IMEX RK2 reference on N = 3200; the reported runs use CFL 0.9 and τ = 10^-8, 0.02, and 1.Source: §3.3.4 · p. 21, Figure 10 · p. 21, Figure 11 · p. 22

3.3.5. 2D Jin-Xin relaxation model.

  • The smooth, well-prepared 2D Jin–Xin case compares numerical and analytic Fourier-transform solutions for u, v, and w at T = 0.35.The computation uses a 400 × 400 Cartesian grid, CFL 0.9, and τ = 10^-10.Source: §3.3.5 · pp. 21, 23, Figure 14 · p. 24
  • Tables 12–14 report experimentally observed L1-errors and convergence orders for u, v, and w in the 2D Jin-Xin model.The results use CFL 0.9, τ = 10^-10, and initial condition (3.45).Source: §3.3.5 · p. 23, Table 12 · p. 24, Table 13 · p. 24
  • For every mesh resolution tested, CS-EBT2, UCS2, and IMEX RK2 preserve the 2D Jin–Xin integral invariant to nearly machine precision.The invariant is assessed through the L1-error in u + τv + τw relative to its initial value, with CFL 0.9, τ = 10^-10, a = 0.7, and b = 0.5.Source: §3.3.5 · pp. 23–25, Table 15 · p. 25

4. conclusion

The paper establishes integral invariants analytically for proposed relaxation formulations of several one- and two-dimensional systems. Numerical experiments verify invariant preservation and second-order accuracy, and indicate advantages of CS-EBT2 for stiff relaxation problems.

4. conclusion

  • The work constructs relaxation formulations for shallow-water, Broadwell, and Euler equations with heat transfer, extends the framework to two-dimensional Jin–Xin, and analytically establishes integral invariants for all considered models.This includes the invariant for the proposed two-dimensional extension.Source: §4 · p. 25
  • The computational verification uses CS-EBT2, UCS2, and IMEX-RK2; CS-EBT2 treats source terms implicitly, while UCS2 combines predictor-stage evaluations with implicit treatment of stiff contributions.IMEX-RK2 is included as an established stiff-source method and generates refined-mesh reference solutions when exact solutions are unavailable.Source: §4 · p. 25
  • Numerical experiments confirm invariant preservation and expected second-order accuracy; compared with UCS2, CS-EBT2 offers a larger stability region, less restrictive CFL conditions, fewer predictor stages, and lower computational cost.These properties make CS-EBT2 particularly attractive for stiff relaxation problems, and the schemes are applicable to the proposed two-dimensional extension.Source: §4 · pp. 25–26
  • The experiments provide theoretical and numerical evidence for the invariant-preserving character of the relaxation formulations across the one-dimensional models and two-dimensional Jin–Xin model.The numerical schemes are demonstrated to be effectively applicable to these models, including the proposed two-dimensional extension.Source: §4 · pp. 25–26

Funding

Both authors’ work is supported by an NBHM, DAE, India project.

Funding

  • Both authors’ work is supported by project NBHM, DAE, India, Ref. No. 02011/46/2021 NBHM(R.P.)/R&D II/14874.Source: Funding · p. 26
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