arXiv subjects

Mohammad R. Najafian

Publications and source records attributed to Mohammad R. Najafian.

3 recordsLinked to original sources

arXiv2026

On the Strong Stability Preserving Property of Runge-Kutta Methods for Hyperbolic Problems

Strong Stability Preserving (SSP) time integration schemes maintain stability of the forward Euler method for any initial value problem. However, only a small subset of Runge-Kutta (RK) methods are SSP, and many efficient high-order time integration schemes do not formally belong to this class. In this work, we introduce a mathematical strategy to analyze the nonlinear stability of RK schemes that may not necessarily belong to the SSP class. With this approach, we mathematically demonstrate that there are time integration schemes outside the class of SSP schemes that can maintain entropy stability and positivity of density and pressure for the Lax-Friedrichs discretization, and Total Variation Diminishing stability for the first-order upwind and the second-order MUSCL schemes. As a result, for these problems, a broader range of RK methods, including the classical fourth-order, four-stage RK scheme, can be used while the numerical integration remains stable. Numerical experiments confirm these theoretical findings, and additional experiments demonstrate similar observations for a wider class of space discretizatins.

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arXiv2024

Quasi-Orthogonal Runge-Kutta Projection Methods

A wide range of physical phenomena exhibit auxiliary admissibility criteria, such as conservation of entropy or various energies, which arise implicitly under exact solution of their governing PDEs. However, standard temporal schemes, such as classical Runge-Kutta (RK) methods, do not enforce these constraints, leading to a loss of accuracy and stability. Projection is an efficient way to address this shortcoming by correcting the RK solution at the end of each time step. Here we introduce a novel projection method for explicit RK schemes, called a \textit{quasi-orthogonal} projection method. This method can be employed for systems containing a single (not necessarily convex) invariant functional, for dissipative systems, and for the systems containing multiple invariants. It works by projecting the orthogonal search direction(s) into the solution space spanned by the RK stage derivatives. With this approach linear invariants of the problem are preserved, the time step size remains fixed, additional computational cost is minimal, and these optimal search direction(s) preserve the order of accuracy of the base RK method. This presents significant advantages over existing projection methods. Numerical results demonstrate that these properties are observed in practice for a range of applications.

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arXiv2024

Energy Conservative Relaxation-Free Runge-Kutta Schemes

A wide range of physical phenomena exhibit auxiliary admissibility criteria, such as conservation of entropy or various energies, which arise implicitly under the exact solution of their governing PDEs. However, standard temporal schemes, such as classical Runge-Kutta (RK) methods, do not enforce these constraints, leading to a loss of accuracy and stability. Previously, the Incremental Directional Technique RK (IDT-RK) and Relaxation Runge-Kutta (R-RK) approaches have been proposed to address this. However, these lead to a loss of accuracy in the case of IDT-RK, or a loss of step size control in the case of R-RK. In the current work we propose Relaxation-Free Runge- Kutta (RF-RK) schemes, which conserve energy, maintain order of accuracy, and maintain a constant step size, alleviating many of the limitations of the aforementioned techniques. Importantly, they do so with minimal additional computational cost compared to the base RK scheme. Numerical results demonstrate that these properties are observed in practice for a range of applications. Therefore, the proposed RF-RK framework is a promising approach for energy conservative time integration of systems of PDEs.

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