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Semi-Implicit Central scheme for Hyperbolic Systems of Balance Laws with Relaxed Source Term

Sudipta Sahu, Emanuele Macca, Rathan Samala

TL;DR

This work develops CS-EBT2, a semi-implicit central scheme for one-dimensional hyperbolic systems of balance laws with stiff relaxation terms. By integrating a Nessyahu–Tadmor predictor–corrector with a semi-implicit trapezoidal update of the relaxed source term, the method achieves asymptotic-preserving behavior and second-order accuracy in the stiff limit while allowing significantly larger CFL numbers than explicit schemes. The approach is validated on Jin-Xin, shallow water, Broadwell, and Euler systems with stiff heat transfer and stiff friction, showing accurate shock resolution, robust performance under stiffness, and favorable comparisons to IMEX-RK2 references. The results indicate strong potential for efficient, robust simulations of stiff hyperbolic dynamics, with avenues for extension to higher dimensions, adaptivity, and positivity-preserving schemes.

Abstract

Quasi-linear hyperbolic systems with source terms introduce significant computational challenges due to the presence of a stiff source term. To address this, a finite volume Nessyahu-Tadmor (NT) central numerical scheme is explored and applied to benchmark models such as the Jin-Xin relaxation model, the shallow-water model, the Broadwell model, the Euler equations with heat transfer, and the Euler system with stiff friction to assess their effectiveness. The core part of this numerical scheme lies in developing a new implicit-explicit (IMEX) scheme, where the stiff source term is handled in an semi-implicit manner constructed by combining the midpoint rule in space, the trapezoidal rule in time with a backward semi-implicit Taylor expansion. The advantage of the proposed method lies in its stability region and maintains robustness near stiffness and discontinuities, while asymptotically preserving second-order accuracy. Theoretical analysis and numerical validation confirm the stability and accuracy of the method, highlighting its potential for efficiently solving the stiff hyperbolic systems of balance laws.

Semi-Implicit Central scheme for Hyperbolic Systems of Balance Laws with Relaxed Source Term

TL;DR

This work develops CS-EBT2, a semi-implicit central scheme for one-dimensional hyperbolic systems of balance laws with stiff relaxation terms. By integrating a Nessyahu–Tadmor predictor–corrector with a semi-implicit trapezoidal update of the relaxed source term, the method achieves asymptotic-preserving behavior and second-order accuracy in the stiff limit while allowing significantly larger CFL numbers than explicit schemes. The approach is validated on Jin-Xin, shallow water, Broadwell, and Euler systems with stiff heat transfer and stiff friction, showing accurate shock resolution, robust performance under stiffness, and favorable comparisons to IMEX-RK2 references. The results indicate strong potential for efficient, robust simulations of stiff hyperbolic dynamics, with avenues for extension to higher dimensions, adaptivity, and positivity-preserving schemes.

Abstract

Quasi-linear hyperbolic systems with source terms introduce significant computational challenges due to the presence of a stiff source term. To address this, a finite volume Nessyahu-Tadmor (NT) central numerical scheme is explored and applied to benchmark models such as the Jin-Xin relaxation model, the shallow-water model, the Broadwell model, the Euler equations with heat transfer, and the Euler system with stiff friction to assess their effectiveness. The core part of this numerical scheme lies in developing a new implicit-explicit (IMEX) scheme, where the stiff source term is handled in an semi-implicit manner constructed by combining the midpoint rule in space, the trapezoidal rule in time with a backward semi-implicit Taylor expansion. The advantage of the proposed method lies in its stability region and maintains robustness near stiffness and discontinuities, while asymptotically preserving second-order accuracy. Theoretical analysis and numerical validation confirm the stability and accuracy of the method, highlighting its potential for efficiently solving the stiff hyperbolic systems of balance laws.

Paper Structure

This paper contains 22 sections, 1 theorem, 99 equations, 16 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Jin-Xin Model
  4. Shallow water Model
  5. Broadwell Model
  6. Euler Equations with Heat Transfer Model
  7. Euler system with stiff friction
  8. Methodology
  9. The Nessyahu--Tadmor predictor–corrector scheme
  10. Approximation of spacetime integral of relaxed source term
  11. Analysis of the scheme CS-EBT2
  12. Consistency analysis.
  13. Stability Analysis
  14. Numerical Examples
  15. Jin-Xin Model
  16. ...and 7 more sections

Key Result

Lemma 4.4

Following Liotta, for any fixed value of $z_{1} \in \mathbb{C}$, the modulus of the function $\Phi\left(z_{1}, \, z_{2}\right)$ assumes its maximum value in the complex half plane $\mathbb{C}^-$ for some $z_2$ belonging to the imaginary axis.

Figures (16)

  • Figure 1: Stability region of the proposed scheme for $\Phi(0,z_2)$ (left) and $S_1$ (right).
  • Figure 2: Jin-Xin model with smooth case: comparison between Numerical solution $u$(left) and $v$(right) and exact solution with CFL $1/3$ and $N=320$.
  • Figure 3: Jin-Xin model with smooth case: comparison between Numerical solution $u$(left) and $v$(right) and exact solution with CFL $0.9$ and $N=320$.
  • Figure 4: Convergence rates for the numerical solutions $u$ and $v$ in the smooth case of the Jin–Xin model, computed on the spatial domain $[0,1]$ up to final time $T=0.35$, using a CFL number of $0.9$ and relaxation parameters $\varepsilon$ ranging from $10^{-6}$ to $10^{-14}$.
  • Figure 5: Jin-Xin model with an unprepared smooth case: comparison between Numerical solution $u$(left) and $v$(right) and exact solution with CFL $0.9$ and $N=320$.
  • ...and 11 more figures

Theorems & Definitions (5)

  • Remark 3.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Lemma 4.4