Related papers: Bound-Preserving Framework for Central-Upwind Sche…
Finite volume schemes for hyperbolic balance laws require a piecewise polynomial reconstruction of the cell averaged values, and a reconstruction is termed `well-balanced' if it is able to simulate steady states at higher order than time…
The paper focuses on first-order invariant-domain preserving approximations of hyperbolic systems. We propose a new way to estimate the artificial viscosity that has to be added to make explicit, conservative, consistent numerical methods…
Kinetic or Boltzmann schemes are interesting alternatives to the macroscopic numerical methods for solving the hyperbolic conservation laws of gas dynamics. They utilize the particle-based description instead of the wave propagation models.…
This paper presents the construction of two numerical schemes for the solution of hyperbolic systems with relaxation source terms. The methods are built by considering the relaxation system as a whole, without separating the resolution of…
The active flux (AF) method is a compact high-order finite volume method that evolves cell averages and point values at cell interfaces independently. Within the method of lines framework, the point value can be updated based on Jacobian…
High-order entropy stable summation-by-parts (SBP) schemes are a class of robust and accurate numerical methods for hyperbolic conservation laws that are numerically stable at arbitrary order without the need for artificial stabilization.…
The constrained transport (CT) method reflects the state of the art numerical technique for preserving the divergence-free condition of magnetic field to machine accuracy in multi-dimensional MHD simulations performed with Godunov-type, or…
We report on the development of a computational framework for the parallel, mesh-adaptive solution of systems of hyperbolic conservation laws like the time-dependent Euler equations in compressible gas dynamics or Magneto-Hydrodynamics…
The paper focuses on the development of numerical methods for the compressible Euler equations. It is well-known that if the Mach number is small, the system becomes stiff and hence explicit schemes suffer from severe time-step…
This paper explores numerical schemes for Temple-class systems, which are integral to various applications including one-dimensional two-phase flow, elasticity, traffic flow, and sedimentation. Temple-class systems are characterized by…
Numerical simulation of the spherically symmetric Einstein--Euler (EE) system faces severe challenges due to the stringent physical admissibility constraints of relativistic fluids and the geometric singularities inherent in metric…
A third-order weighted essentially non-oscillatory compact least-squares scheme is developed for the finite volume method on structured curvilinear non-uniform grids. The proposed scheme features compact least-squares reconstruction with…
In this paper, we construct a robust adaptive central-upwind scheme on unstructured triangular grids for two-dimensional shallow water equations with variable density. The method is well-balanced, positivity-preserving, and oscillation-free…
We study some theoretical aspects of $P_NP_M$ schemes, which are a novel class of high order accurate reconstruction based discontinuous Galerkin (DG) schemes for hyperbolic conservation laws. The PNPM schemes store and evolve the discrete…
We present a new third-order central scheme for approximating solutions of systems of conservation laws in one and two space dimensions. In the spirit of Godunov-type schemes,our method is based on reconstructing a piecewise-polynomial…
Machine learning techniques are being used as an alternative to traditional numerical discretization methods for solving hyperbolic partial differential equations (PDEs) relevant to fluid flow. Whilst numerical methods are higher fidelity,…
A system of hyperbolic conservation laws $$ \partial_t u + \partial_x \partial_u Q = 0, \quad Q = u_1^3 / 3 + u_1 u_2^2, \qquad u = u(x,t) \in\mR^2, $$ as well as its viscous regularization $$ \partial_t u + \partial_x \partial_u Q = \calM…
Many entropy-conservative and entropy-stable (summarized as entropy-preserving) methods for hyperbolic conservation laws rely on Tadmor's theory for two-point entropy-preserving numerical fluxes and its higher-order extension via flux…
Polynomial spectral methods produce fast, accurate, and flexible solvers for broad ranges of PDEs with one bounded dimension, where the incorporation of general boundary conditions is well understood. However, automating extensions to…
Physical solutions to the widely used Aw-Rascle-Zhang (ARZ) traffic model and the adapted pressure (AP) ARZ model should satisfy the positivity of density, the minimum and maximum principles with respect to the velocity $v$ and other…