arXiv subjects

K. R. Arun

Publications and source records attributed to K. R. Arun.

18 recordsLinked to original sources

arXiv2026

Error analysis of an asymptotic-preserving, energy-stable finite volume method for barotropic Euler equations

We propose and analyse an energy-stable and asymptotic-preserving finite volume scheme for the compressible Euler system. Using the relative energy framework, we establish rigorous error estimates that yield convergence of the numerical solutions in two distinct regimes. For a fixed Mach number $\varepsilon>0$, we derive error estimates between the numerical solutions and a strong solution of the compressible Euler system that are uniform with respect to the discretisation parameters, ensuring convergence as the underlying mesh is refined. In the low Mach number regime, we analyse the error between the numerical solutions and a strong solution of the incompressible Euler system and obtain asymptotic error estimates that are uniform in $\varepsilon$ and the discretisation parameters. These results imply convergence of the numerical solutions toward a strong solution of the incompressible Euler system as $\varepsilon$, and the discretisation parameters simultaneously tend to zero. Numerical experiments are presented to validate the theoretical analysis.

math.NA↗

arXiv2025

A Convergent Structure-Preserving Scheme for Dissipative Solutions of the Rotating Shallow Water System

We design and analyse a semi-implicit finite volume scheme for the two-dimensional rotating shallow water (RSW) equations that is energy stable, well-balanced (capable of preserving discrete geostrophic steady states), consistent, and covergent. The key idea is the introduction of carefully chosen stabilisation terms into the convective fluxes of the mass and momentum equations, as well as the source terms. Under a CFL-type condition, together with an auxiliary time-step restriction arising from the Coriolis forces, we establish the energy stability of the scheme. The stabilisation terms are constructed to vanish at steady states, thereby ensuring the well-balancing property under an appropriate advective CFL condition. We derive a sufficient time-step restriction that guarantees stability, well-balancing, existence of discrete solutions, and positivity simultaneously. Furthermore, under mild boundedness assumptions, we obtain a priori estimates showing that the stabilisation terms converge to zero as the mesh is refined, which establishes the consistency of the scheme. This in turn enables us to prove that numerical solutions generate a Young measure, identifiable as a dissipative measure-valued solution of the RSW system, thereby yielding convergence of the scheme. Finally, we confirm the theoretical results through extensive numerical experiments.

math.NA↗

arXiv2025

A Structure-Preserving Scheme for the Euler System with Potential Temperature Transport

We consider the compressible Euler equations with potential temperature transport, a system widely used in atmospheric modelling to describe adiabatic, inviscid flows. In the low Mach number regime, the equations become stiff and pose significant numerical challenges. We develop an all-speed, semi-implicit finite volume scheme that is asymptotic preserving (AP) in the low Mach limit and strictly positivity preserving for density and potential temperature. The scheme ensures stability and accuracy across a broad range of Mach numbers, from fully compressible to nearly incompressible regimes. We rigorously establish consistency with both the compressible system and its incompressible, density-dependent limit. Numerical experiments confirm that the method robustly captures complex flow features while preserving the essential physical and mathematical structures of the model.

math.NA↗

arXiv2024

An Energy Stable and Well-balanced Scheme for the Ripa System

We design and analyse an energy stable, structure preserving and well-balanced scheme for the Ripa system of shallow water equations. The energy stability of the numerical solutions is achieved by introducing appropriate stabilisation terms in the discretisation of the convective fluxes of mass and momenta, the pressure gradient and the topography source term. A diligent choice of the interface values of the water height and the temperature ensures the well-balancing property of the scheme for three physically relevant hydrostatic steady states. The explicit in time and finite volume in space scheme preserves the positivity of the water height and the temperature, and it is weakly consistent with the continuous model equations in the sense of Lax-Wendroff. The results of extensive numerical case studies on benchmark test problems are presented to confirm the theoretical findings.

math.NA↗

arXiv2024

An Asymptotic Preserving Scheme for the Euler-Poisson-Boltzmann System in the Quasineutral Limit

In this paper, we study an asymptotic preserving (AP), energy stable and positivity preserving semi-implicit finite volume scheme for the Euler-Poisson-Boltzmann (EPB) system in the quasineutral limit. The key to energy stability is the addition of appropriate stabilisation terms into the convective fluxes of mass and momenta, and the source term. The space-time fully-discrete scheme admits the positivity of the mass density, and is consistent with the weak formulation of the EPB system upon mesh refinement. In the quasineutral limit, the numerical scheme yields a consistent, semi-implicit discretisation of the isothermal compressible Euler system, thus leading to the AP property. Several benchmark numerical case studies are performed to confirm the robustness and efficacy of the proposed scheme in the dispersive as well as the quasineutral regimes. The numerical results also corroborates scheme's ability to very well resolve plasma sheaths and the related dynamics, which indicates its potential to applications involving low-temperature plasma problems.

math.NA↗

arXiv2024

Asymptotic preserving finite volume method for the compressible Euler equations: analysis via dissipative measure-valued solutions

We propose and analyze a new asymptotic preserving (AP) finite volume scheme for the multidimensional compressible barotropic Euler equations to simulate low Mach number flows. The proposed scheme uses a stabilized upwind numerical flux, with the stabilization term being proportional to the stiff pressure gradient, and we prove its conditional energy stability and consistency. Utilizing the concept of dissipative measure-valued (DMV) solutions, we rigorously illustrate the AP properties of the scheme for well-prepared initial data. In particular, we prove that the numerical solutions will converge weakly to a DMV solution of the compressible Euler equations as the mesh parameter vanishes, while the Mach number is fixed. The DMV solutions then converge to a classical solution of the incompressible Euler system as the Mach number goes to zero. Conversely, we show that if the mesh parameter is kept fixed, we obtain an energy stable and consistent finite-volume scheme approximating the incompressible Euler equations as the Mach number goes to zero. The numerical solutions generated by this scheme then converge weakly to a DMV solution of the incompressible Euler system as the mesh parameter vanishes. Invoking the weak-strong uniqueness principle, we conclude that the DMV solution and classical solution of the incompressible Euler system coincide, proving the AP property of the scheme. We also present an extensive numerical case study in order to illustrate the theoretical convergences, wherein we utilize the techniques of K-convergence.

math.NA↗

arXiv2024

An Asymptotic Preserving and Energy Stable Scheme for the Euler System with Congestion Constraint

In this work, we design and analyze an asymptotic preserving (AP), semi-implicit finite volume scheme for the scaled compressible isentropic Euler system with a singular pressure law known as the congestion pressure law. The congestion pressure law imposes a maximal density constraint of the form $0\leq \varrho <1$, and the scaling introduces a small parameter $\varepsilon$ in order to control the stiffness of the density constraint. As $\varepsilon\to 0$, the solutions of the compressible system converge to solutions of the so-called free-congested Euler equations that couples compressible and incompressible dynamics. We show that the proposed scheme is positivity preserving and energy stable. In addition, we also show that the numerical densities satisfy a discrete variant of the constraint. By means of extensive numerical case studies, we verify the efficacy of the scheme and show that the scheme is able to capture the two dynamics in the limiting regime, thereby proving the AP property.

math.NA↗

arXiv2024

An Energy Stable Well-balanced Scheme for the Barotropic Euler System with Gravity under the Anelastic Scaling

We design and analyse an energy stable, structure preserving, well-balanced and asymptotic preserving (AP) scheme for the barotropic Euler system with gravity in the anelastic limit. The key to energy stability is the introduction of appropriate velocity shifts in the convective fluxes of mass and momenta. The semi-implicit in time and finite volume in space fully-discrete scheme supports the positivity of density and yields the consistency with the weak solutions of the Euler system upon mesh refinement. The numerical scheme admits the discrete hydrostatic states as solutions and the stability of numerical solutions in terms of the relative energy leads to well-balancing. The AP property of the scheme, i.e. the boundedness of the mesh parameters with respect to the Mach/Froude numbers and the scheme's asymptotic consistency with the anelastic Euler system is rigorously shown on the basis of apriori energy estimates. The numerical scheme is resolved in two steps: by solving a non-linear elliptic problem for the density and a subsequent explicit computation of the velocity. Results from several benchmark case studies are presented to corroborate the proposed claims.

math.NA↗

arXiv2023

A semi-implicit finite volume scheme for dissipative measure-valued solutions to the barotropic Euler system

A semi-implicit in time, entropy stable finite volume scheme for the compressible barotropic Euler system is designed and analyzed and its weak convergence to a dissipative measure-valued (DMV) solution [E. Feireisl et al., Dissipative measure-valued solutions to the compressible Navier-Stokes system, Calc. Var. Partial Differential Equations, 2016] of the Euler system is shown. The entropy stability is achieved by introducing a shifted velocity in the convective fluxes of the mass and momentum balances, provided some CFL-like condition is satisfied to ensure stability. A consistency analysis is performed in the spirit of the Lax's equivalence theorem under some physically reasonable boundedness assumptions. The concept of K-convergence [E. Feireisl et al., K-convergence as a new tool in numerical analysis, IMA J. Numer. Anal., 2020] is used in order to obtain some strong convergence results, which are then illustrated via rigorous numerical case studies. The convergence of the scheme to a DMV solution, a weak solution and a strong solution of the Euler system using the weak-strong uniqueness principle and relative entropy are presented.

math.NA↗

arXiv2023

An Asymptotic Preserving and Energy Stable Scheme for the Euler-Poisson System in the Quasineutral Limit

An asymptotic preserving and energy stable scheme for the Euler-Poisson system under the quasineutral scaling is designed and analysed. Correction terms are introduced in the convective fluxes and the electrostatic potential, which lead to the dissipation of mechanical energy and the entropy stability. The resolution of the semi-implicit in time finite volume in space fully-discrete scheme involves two steps: the solution of an elliptic problem for the potential and an explicit evaluation for the density and velocity. The proposed scheme possesses several physically relevant attributes, such as the the entropy stability and the consistency with the weak formulation of the continuous Euler-Poisson system. The AP property of the scheme, i.e. the boundedness of the mesh parameters with respect to the Debye length and its consistency with the quasineutral limit system, is shown. The results of numerical case studies are presented to substantiate the robustness and efficiency of the proposed method.

math.NA↗

arXiv2023

An Asymptotic Preserving and Energy Stable Scheme for the Barotropic Euler System in the Incompressible Limit

An asymptotic preserving and energy stable scheme for the barotropic Euler system under the low Mach number scaling is designed and analysed. A velocity shift proportional to the pressure gradient is introduced in the convective fluxes, which leads to the dissipation of mechanical energy and the entropy stability at all Mach numbers. The resolution of the semi-implicit in time and upwind in space fully-discrete scheme involves two steps: solution of an elliptic problem for the density and an explicit evaluation for the velocity. The proposed scheme possess several physically relevant attributes, such as the positivity of density, the entropy stability and the consistency with the weak formulation of the continuous Euler system. The AP property of the scheme, i.e.\ the boundedness of the mesh parameters with respect to the Mach number and its consistency with the incompressible limit system, is shown rigorously. The results of extensive case studies are presented to substantiate the robustness and efficacy of the proposed scheme as well as the theoretical claims.

math.NA↗

arXiv2022

High Order Asymptotic Preserving and Classical Semi-implicit RK Schemes for the Euler-Poisson System in the Quasineutral Limit

In this paper, the design and analysis of high order accurate IMEX finite volume schemes for the compressible Euler-Poisson (EP) equations in the quasineutral limit is presented. As the quasineutral limit is singular for the governing equations, the time discretisation is tantamount to achieving an accurate numerical method. To this end, the EP system is viewed as a differential algebraic equation system (DAEs) via the method of lines. As a consequence of this vantage point, high order linearly semi-implicit (SI) time discretisation are realised by employing a novel combination of the direct approach used for implicit discretisation of DAEs and, two different classes of IMEX-RK schemes: the additive and the multiplicative. For both the time discretisation strategies, in order to account for rapid plasma oscillations in quasineutral regimes, the nonlinear Euler fluxes are split into two different combinations of stiff and non-stiff components. The high order scheme resulting from the additive approach is designated as a classical scheme while the one generated by the multiplicative approach possesses the asymptotic preserving (AP) property. Time discretisations for the classical and the AP schemes are performed by standard IMEX-RK and SI-IMEX-RK methods, respectively so that the stiff terms are treated implicitly and the non-stiff ones explicitly. In order to discretise in space a Rusanov-type central flux is used for the non-stiff part, and simple central differencing for the stiff part. AP property is also established for the space-time fully-discrete scheme obtained using the multiplicative approach. Results of numerical experiments are presented, which confirm that the high order schemes based on the SI-IMEX-RK time discretisation achieve uniform second order convergence with respect to the Debye length and are AP in the quasineutral limit.

math.NA↗

arXiv2022

System of kinematical conservation laws (KCL) a review article

In a wide range of physical phenomena, we find propagating surfaces Ωt which need mathematical treatment. In this article, we review the theory of the system of kinematical conservation laws (KCL), which govern the evolution of these surfaces and have been developed by the second author and his collaborators. KCL are the most general equations in conservation form, governing the evolution of Ωt with physically realistic singularities. A very special type of singularity is a kink, which is a point on Ωt when Ωt is a curve in R2 andisacurveonΩt whenΩt isasurfaceinR3. AcrossakinkthenormalntoΩt and the normal velocity m on Ωt are discontinuous. The main aim of this article is to identify density of the conserved variable and the flux for the KCL which we did not do earlier. The presentation of this article is like that in a popular article, which which aims at non-experts in the field.

math.AP↗

arXiv2021

A Unified Asymptotic Preserving and Well-balanced Scheme for the Euler System with Multiscale Relaxation

The design and analysis of a unified asymptotic preserving (AP) and well-balanced scheme for the Euler Equations with gravitational and frictional source terms is presented in this paper. The asymptotic behaviour of the Euler system in the limit of zero Mach and Froude numbers, and large friction is characterised by an additional scaling parameter. Depending on the values of this parameter, the Euler system relaxes towards a hyperbolic or a parabolic limit equation. Standard Implicit-Explicit Runge-Kutta schemes are incapable of switching between these asymptotic regimes. We propose a time semi-discretisation to obtain a unified scheme which is AP for the two different limits. A further reformulation of the semi-implicit scheme can be recast as a fully-explicit method in which the mass update contains both hyperbolic and parabolic fluxes. A space-time fully-discrete scheme is derived using a finite volume framework. A hydrostatic reconstruction strategy, an upwinding of the sources at the interfaces, and a careful choice of the central discretisation of the parabolic fluxes are used to achieve the well-balancing property for hydrostatic steady states. Results of several numerical case studies are presented to substantiate the theoretical claims and to verify the robustness of the scheme.

math.NA↗

arXiv2019

Asymptotic Preserving and Low Mach Number Accurate IMEX Finite Volume Schemes for the Isentropic Euler Equations

In this paper, the design and analysis of a class of second order accurate IMEX finite volume schemes for the compressible Euler equations in the zero Mach number limit is presented. In order to account for the fast and slow waves, the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components, respectively. The time discretisation is performed by an IMEX Runge-Kutta method, therein the stiff terms are treated implicitly and the non-stiff terms explicitly. In the space discretisation, a Rusanov-type central flux is used for the non-stiff part, and simple central differencing for the stiff part. Both the time semi-discrete and space-time fully-discrete schemes are shown to be asymptotic preserving. The numerical experiments confirm that the schemes achieve uniform second order convergence with respect to the Mach number. A notion of accuracy at low Mach numbers, termed as the asymptotic accuracy, is introduced in terms of the invariance of a well-prepared space of constant densities and divergence-free velocities. The asymptotic accuracy is concerned with the closeness of the compressible solution with that of its incompressible counterpart in a low Mach number regime. It is shown theoretically as well as numerically that the proposed schemes are asymptotically accurate.

math.NA↗

arXiv2019

An Asymptotic Preserving Time Integrator for Low Mach Number Limits of the Euler Equations with Gravity

We consider two distinguished asymptotic limits of the Euler equations in a gravitational field, namely the incompressible and Boussinesq limits. Both these limits can be obtained as singular limits of the Euler equations under appropriate scaling of the Mach and Froude numbers. We propose and analyse an asymptotic preserving (AP) time discretisation for the numerical approximation of the Euler system in these asymptotic regimes. A key step in the construction of the AP scheme is a semi-implicit discretisation of the fluxes and the source term. The non-stiff convective terms are treated explicitly whereas the stiff pressure-gradient and source term are implicit. The implicit terms are combined to get a nonlinear elliptic equation. We show that the overall scheme is consistent with the respective limit system when the Mach number goes to zero. A linearised stability analysis confirms the $L^2$-stability of the proposed scheme. The results of numerical experiments validate the theoretical findings.

math.NA↗

arXiv2017

Propagation of a Three-dimensional Weak Shock Front Using Kinematical Conservation Laws

In this paper we present a mathematical theory and a numerical method to study the propagation of a three-dimensional (3-D) weak shock front into a polytropic gas in a uniform state and at rest, though the method can be extended to shocks moving into nonuniform flows. The theory is based on the use of 3-D kinematical conservation laws (KCL), which govern the evolution of a surface in general and a shock front in particular. The 3-D KCL, derived purely on geometrical considerations, form an under-determined system of conservation laws. In the present paper the 3-D KCL system is closed by using two appropriately truncated transport equations from an infinite hierarchy of compatibility conditions along shock rays. The resulting governing equations of this KCL based 3-D shock ray theory, leads to a weakly hyperbolic system of eight conservation laws with three divergence-free constraints. The conservation laws are solved using a Godunov-type central finite volume scheme, with a constrained transport technique to enforce the constraints. The results of extensive numerical simulations reveal several physically realistic geometrical features of shock fronts and the complex structures of kink lines formed on them. A comparison of the results with those of a weakly nonlinear wavefront shows that a weak shock front and a weakly nonlinear wavefront are topologically same. The major important differences between the two are highlighted in the contexts of corrugational stability and converging shock fronts.

math.AP↗

arXiv2014

A Weakly Asymptotic Preserving Low Mach Number Scheme for the Euler Equations of Gas Dynamics

We propose a low Mach number, Godunov-type finite volume scheme for the numerical solution of the compressible Euler equations of gas dynamics. The scheme combines Klein's non-stiff/stiff decomposition of the fluxes (J. Comput. Phys. 121:213-237, 1995) with an explicit/implicit time discretization (Cordier et al., J. Comput. Phys. 231:5685- 5704, 2012) for the split fluxes. This results in a scalar second order partial differential equation (PDE) for the pressure, which we solve by an iterative approximation. Due to our choice of a crucial reference pressure, the stiff subsystem is hyperbolic, and the second order PDE for the pressure is elliptic. The scheme is also uniformly asymptotically consistent. Numerical experiments show that the scheme needs to be stabilized for low Mach numbers. Unfortunately, this affects the asymptotic consistency, which becomes non-uniform in the Mach number, and requires an unduly fine grid in the small Mach number limit. On the other hand, the CFL number is only related to the non-stiff characteristic speeds, independently of the Mach number. Our analytical and numerical results stress the importance of further studies of asymptotic stability in the development of AP (asymptotic preserving) schemes.

math.NA↗