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Efficient Hybrid WENO Schemes for Special Relativistic Hydrodynamics with Adaptive Characteristic Reconstruction
Rakesh Kumar, Biswarup Biswas, Asha Kumari Meena, Harish Kumar
TL;DR
SRHD simulations require stable, accurate treatment of discontinuities, but component-wise WENO can oscillate while characteristic-wise WENO is computationally expensive. The proposed H1-WENO, H2-WENO, and H3-WENO schemes use a troubled-cell indicator to apply inexpensive reconstructions in smooth regions and characteristic-wise WENO near discontinuities, achieving an effective balance between computational efficiency and numerical accuracy.
Problem
Component-wise WENO may produce spurious oscillations near discontinuities, whereas characteristic-wise WENO improves stability but substantially increases computational cost.
Method
The H1-WENO, H2-WENO, and H3-WENO schemes use a smoothness-indicator-based troubled-cell detector to adaptively switch reconstruction strategies across smooth and non-smooth regions.
Results
Numerical experiments show that the hybrid schemes retain characteristic-wise WENO's accuracy and non-oscillatory behavior while reducing computational cost.
Takeaways & Limitations
The hybrid schemes provide an effective balance between computational efficiency and numerical accuracy for large-scale simulations involving complex hyperbolic conservation-law systems.
Takeaways & Limitations
The formulation assumes units with c = 1 and adopts an ideal relativistic equation of state.
Abstract
from arXiv · showhide
Special relativistic hydrodynamics (SRHD) equations arise in the modeling of high-speed fluid flows encountered in astrophysical phenomena such as jets, supernova explosions, and gamma-ray bursts. Owing to their highly nonlinear hyperbolic nature, solutions often develop strong discontinuities, making the design of stable and accurate numerical schemes challenging. Although Weighted Essentially Non-Oscillatory (WENO) schemes are widely used for such problems, component-wise WENO reconstruction may produce spurious oscillations near discontinuities. On the other hand, characteristic-wise WENO reconstruction provides accurate non-oscillatory solutions for systems of conservation laws, but it involves the computation of eigenvectors in each cell, which leads to high computational cost. In this work, we intend to develop hybrid schemes which maintain the non-oscillatory feature of characteristic-wise WENO while being less costly. We propose three hybrid schemes, namely the H1-WENO, H2-WENO, and H3-WENO schemes, based on a new troubled-cell indicator constructed from the smoothness indicators of the WENO scheme. The proposed troubled-cell indicator effectively distinguishes smooth and discontinuous regions, allowing the hybrid schemes to employ inexpensive reconstructions in smooth regions and the characteristic-wise WENO reconstruction only near discontinuities. Numerical experiments demonstrate that the proposed schemes retain the accuracy and robustness of characteristic-wise WENO methods while significantly reducing the computational cost. In particular, the H1-WENO scheme achieves an approximately 30--40% improvement in computational efficiency compared to the standard WENO scheme in 2D test cases.
1. Introduction
SRHD involves highly nonlinear hyperbolic conservation laws whose discontinuous solutions make stable, accurate, non-oscillatory computation difficult. The paper proposes adaptive hybrid WENO schemes to reduce characteristic-wise reconstruction cost while preserving its numerical advantages.
- Motivation: SRHD models relativistic flows in applications including astrophysical jets, supernova explosions, gamma-ray bursts, accretion disks, and high-energy plasma flows.The governing equations arise from conservation of mass, momentum, and energy in special relativity.
- Motivation: The highly nonlinear hyperbolic equations can develop shocks and contact discontinuities, requiring robust high-resolution shock-capturing schemes.Numerical methods must avoid spurious oscillations and capture complex wave interactions accurately.
- Existing WENO approaches: Component-wise WENO may generate spurious oscillations near discontinuities because of nonlinear coupling among characteristic fields.Characteristic-wise reconstruction improves stability by projecting variables onto local characteristic fields, but requires eigenvectors and increases computational cost.
- Proposed approach: The paper develops H1-WENO, H2-WENO, and H3-WENO schemes using a troubled-cell indicator based on WENO smoothness indicators.The schemes adapt reconstruction strategies between smooth and non-smooth regions.
- Validation: The paper validates the proposed schemes through one- and two-dimensional test problems and comparative performance studies.The supplied introduction states the validation scope but does not report detailed numerical outcomes here.
2. Special Relativistic Hydrodynamics Equations
The paper formulates SRHD as conservative hyperbolic equations in one and two dimensions, with relativistic variables, an ideal relativistic equation of state, and Jacobians characterized through eigenvalues and eigenvectors.
- Governing equations: SRHD describes relativistic fluids using conserved mass, momentum, and energy variables in divergence form.The formulation covers spatial dimensions d ∈ {1, 2}.
- Physical variables: The conserved variables use rest-mass density ρ, pressure p, Lorentz factor W, specific enthalpy h, and internal energy e.The Lorentz factor is W = (1 − |u|^2)^−1/2, and h = 1 + e + p/ρ.
- Closure and recovery: The system is closed with an ideal relativistic equation of state, while primitive-variable recovery typically requires solving a nonlinear algebraic equation for pressure.This difficulty follows from nonlinear coupling among the conserved variables.
- Eigenstructure: In one dimension, the flux Jacobian is strictly hyperbolic and admits three distinct real eigenvalues with a complete eigenvector decomposition.The decomposition uses right eigenvectors R, left eigenvectors L = R^−1, and a diagonal eigenvalue matrix D.
- Eigenstructure: In two dimensions, the x- and y-direction Jacobians are separately decomposed into real eigenvalues and complete sets of eigenvectors.The one-dimensional eigenvalues are recovered when the transverse velocity vanishes.
3. Finite Difference Scheme (FDM)
The finite-difference scheme discretizes SRHD conservation laws on uniform cells, splits fluxes using local Lax–Friedrichs formulas, and reconstructs interface fluxes component-wise or characteristic-wise in each spatial direction.
- Finite-difference discretization: The finite-difference formulation represents the conservation law through a semi-discrete evolution of solution values at grid points and numerical flux differences.The spatial domain is divided into non-overlapping control volumes with uniform cell width Δx.
- Flux splitting: The physical flux is decomposed into right-going and left-going components associated with non-negative and nonpositive characteristic speeds.This splitting supports separate nonlinear WENO reconstruction of the flux components.
- Flux splitting: Local Lax–Friedrichs splitting uses a locally computed maximum absolute eigenvalue of the flux Jacobian.The split fluxes are formed from F(U) ± λU in one dimension.
- Reconstruction choices: Component-wise reconstruction applies WENO separately to each flux component, whereas characteristic-wise reconstruction computes interface eigenvectors and reconstructs in characteristic variables.The characteristic-wise procedure uses an average of neighboring primitive variables at the interface.
- Multiple dimensions: The two-dimensional method applies the one-dimensional reconstruction independently in the x and y directions while holding the transverse index fixed.Uniform rectangular cells use separate mesh widths Δx and Δy, with direction-specific maximum eigenvalues λx and λy.
4. Reconstruction Methodology: WENO-AO(5,3)
The paper uses WENO-AO(5,3) for high-order spatial reconstruction, combining a large central polynomial with compact lower-degree polynomials through nonlinear smoothness-dependent weights.
- Reconstruction structure: WENO-AO(5,3) combines one fourth-degree polynomial on a five-point stencil with three quadratic polynomials on compact three-point sub-stencils.The reconstruction is used within the finite-difference framework.
- Accuracy and stability: The method achieves fifth-order accuracy in smooth regions and reduces to third-order accuracy near discontinuities.This adaptive behavior improves stability while retaining high resolution.
- Smoothness indicators: Smoothness indicators are computed for each stencil and polynomial degree to quantify the regularity of candidate reconstructions.The indicators are expressed using squared polynomial-coefficient terms.
- Nonlinear weighting: Nonlinear weights are constructed from the smoothness indicators and corresponding linear weights, then normalized to form the final reconstruction.The implementation uses ϵ = 10^-12 to avoid division by zero.
- Adaptive behavior: In smooth regions, the nonlinear weights approach the optimal linear weights and recover fifth-order accuracy; near discontinuities, nonsmooth stencils are suppressed.The reconstruction then becomes a convex combination of quadratic polynomials with an essentially non-oscillatory third-order profile.
5. Hybrid FDM-WENO Reconstruction
The section develops a troubled-cell indicator from WENO smoothness information and uses it to switch reconstruction strategies between smooth and non-smooth regions. Three hybrid schemes are proposed, differing in their low-cost smooth-region reconstructions while retaining characteristic-wise WENO near discontinuities.
- Component-wise reconstruction is computationally efficient but may introduce spurious small-scale oscillations in hyperbolic systems.
- The troubled-cell indicator is designed to detect discontinuities while avoiding false marking of smooth cells for computational efficiency.A constant K scales the estimate of λ^2Δx^2 to reduce unwanted marking of smooth cells.
- The proposed indicator uses WENO-AO(5,3) smoothness indicators over neighboring five-point stencils and is extended from scalar fluxes to systems.The smooth-flux estimate assumes regularity over both stencils and uses λ = max |f′(u)| over a suitable range.
- H1-WENO uses component-wise WENO in smooth regions and characteristic-wise WENO in non-smooth regions.The scheme switches reconstruction according to the troubled-cell classification.
- H2-WENO uses fifth-order component-wise linear reconstruction in smooth regions, while H3-WENO uses fifth-order characteristic-wise linear reconstruction there.Both schemes apply characteristic-wise WENO reconstruction in non-smooth regions.
- H2-WENO is identified as the most computationally efficient scheme because it avoids eigenvector transformation and nonlinear weights in smooth regions.The numerical computations use K = 20 and K = 40 for 1D and 2D H1-WENO and H2-WENO, while H3-WENO uses K = 1 in both dimensions.
6. Numerical results
The hybrid schemes reproduce WENO-AO solutions across one- and two-dimensional tests while adaptively identifying nonsmooth regions. They preserve accuracy and non-oscillatory behavior, with substantial efficiency gains in smooth problems.
- The third-order TVD Runge–Kutta method advances the semi-discrete system after spatial discretization.
- Accuracy and efficiency: Nearly 50% lower computational time is achieved by H2-WENO for comparable L1-error than WENO-AO in the accuracy test.
- Accuracy and efficiency: H1-WENO improves efficiency by approximately 40%, whereas H3-WENO gains approximately 10% relative to WENO-AO.
- Accuracy and efficiency: All proposed schemes maintain fifth-order convergence, with L2-errors close to one another and no troubled cells detected for the smooth solution.
- One-dimensional tests: The hybrid schemes closely match WENO-AO and reference solutions while resolving wave structures without noticeable spurious oscillations in one-dimensional tests.
- Two-dimensional tests: On two-dimensional tests, the schemes capture complex flow structures and discontinuities, while indicators identify nonsmooth regions and H3-WENO marks fewer unnecessary smooth cells.
7. Conclusion
The proposed H1-WENO, H2-WENO, and H3-WENO schemes use a troubled-cell indicator to balance computational efficiency with numerical accuracy. The indicator’s parameter sensitivity and performance are examined through speed-up, troubled-cell, and solution-profile comparisons.
- The work proposes H1-WENO, H2-WENO, and H3-WENO schemes for efficient numerical approximation of SRHD equations.
- Figure 20 compares speed-up factors and percentages of troubled cells for different K values in 2D tests using H1-WENO and H2-WENO.
- The hybrid schemes use inexpensive reconstructions in smooth regions and characteristic-wise WENO reconstruction near discontinuities.
- Characteristic decomposition, particularly eigenvector computation, is identified as a major computational expense addressed by the hybrid strategy.
- Figure 21 compares troubled-cell indicator plots for K = 1 and K = 100 and the corresponding density profiles on a 500-point grid.
- The troubled-cell indicator does not significantly affect accuracy or computational efficiency when selected within a suitable range.