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The carbuncle phenomenon is a shock instability mechanism which ruins all efforts to compute grid-aligned shock waves using low-dissipative upwind schemes. The present study develops a stability analysis for two-dimensional steady shocks on structured meshes based on the matrix method. The numerical resolution of the corresponding eigenvalue problem confirms the typical odd–even form of the unstable mode and displays a Mach number threshold effect currently observed in computations. Furthermore, the present method indicates that the instability of steady shocks is not only governed by the upstream Mach number but also by the numerical shock structure. Finally, the source of the instability is localized in the upstream region, providing some clues to better understand and control the onset of the carbuncle. [Copyright &y& Elsevier]

Modern shock-capturing schemes often suffer from numerical shock instabilities when simulating strong shocks, limiting their application in supersonic or hypersonic flow simulations. In the current study, we devote our efforts to investigating the shock instability problem for second-order schemes, which has not gotten enough attention in previous research but is crucial to address. To this end, we develop the matrix stability analysis method for the finite-volume Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) approach, taking into account the influence of reconstruction. With the help of this newly developed method, the shock instability of second-order schemes is investigated quantitatively and efficiently. The results demonstrate that when second-order schemes are employed, whether shock instabilities will occur is closely related to the property of Riemann solvers, just like the first-order case. However, enhancing spatial accuracy still impacts the shock instability problem, and the impact can be categorized into two types depending on the dissipation of Riemann solvers. Furthermore, the research emphasizes the impact of the numerical shock structure, highlighting both its role as the source of instability and the influence of its state on the occurrence of instability. One of the most significant contributions of this study is the confirmation of the multidimensional coupled nature of shock instability. Both one-dimensional and multidimensional instabilities are proven to influence the instability problem, and they have different properties. Moreover, this paper reveals that increasing the aspect ratio and distortion angle of the computational grid can help mitigate shock instabilities. The current work provides an effective tool for quantitatively investigating the shock instabilities for second-order schemes, revealing the inherent mechanism and thereby contributing to the elimination of shock instability. [ABSTRACT FROM AUTHOR]

We investigate the stability of a temporal discretization of interface terms in grid overlapping methods. A matrix stability analysis is performed on a model problem of the one-dimensional diffusion equation on overlapping grids. The scheme stability is first analyzed theoretically, and a proof of the unconditional stability of the first-order interface extrapolation scheme with the first- and second-order time integration for any overlap size is presented. For the higher-order schemes, we obtain explicit estimates of the spectral radius of the corresponding discrete matrix operator and document the values of the stability threshold depending on the number of grid points and the size of overlap. The influence of iterations on stability properties is also investigated. Numerical experiments are then presented relating the obtained stability bounds to the observed numerical values. Semidiscrete analysis confirms the derived scaling for the stability bounds. [ABSTRACT FROM AUTHOR]

A systematic procedure is presented to derive stability conditions for leap-frog-type finite-difference schemes for the multidimensional constant-coefficient convective-diffusion equation. The treatise includes von Neumann stability analysis for pure initial-value problems and matrix stability analysis for Dirichlet boundary-value problems. Discrepancies on time-step restriction between von Neumann stability analysis and matrix stability analysis are observed. [ABSTRACT FROM AUTHOR]

The aim of this work is to investigate the asymptotic stability of a viscoelastic Bresse system in one dimensional bounded domain. In this context, we introduce two internal damping terms expressed using the generalized Caputo fractional derivative. By adopting a diffusive representation, we show the well-posedness of the proposed system and we prove some decay results. In order to validate the theoretical findings, we implement a finite difference method and we conduct intensive numerical simulations. Moreover, we provide some insights on the convergence of the elaborated numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper, the authors developed a new differential quadrature method "exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM)” by using exponential modified cubic B-spline functions as test functions in the traditional differential quadrature method [32]. The new method is tested on one and two dimensional nonlinear Burgers’ equations. To check the efficiency and accuracy of the proposed method five numerical problems have been considered. The numerical results of the method are compared with some existing methods and found that the proposed numerical method produces more accurate results than existing methods. Stability analysis of the algorithm is also done by using matrix stability analysis method. [ABSTRACT FROM AUTHOR]

The present paper uses a relatively new approach and methodology to solve second order two dimensional hyperbolic telegraph equation numerically. We use modified cubic B-spline basis functions based differential quadrature method for space discretization that reduces the problem into an amenable system of ordinary differential equations. The resulting system of ODEs in time subsequently have been solved by SSP-RK43 scheme. Stability of the scheme is studied using matrix stability analysis and found to be stable. The efficacy of proposed approach has been confirmed with seven numerical experiments, where comparison is made with some earlier work. It is clear that the results obtained are acceptable and are in good agreement with earlier studies. However, we obtain these results in much less CPU time. The method is very simple, efficient and produces very accurate numerical results in considerably smaller number of nodes and hence saves computational effort. [ABSTRACT FROM AUTHOR]

It is well known that many contact- and shear-preserving approximate Riemann solvers admit various forms of numerical shock anomalies and numerical shock instabilities such as the carbuncle phenomenon, sonic glitch, and post-shock wave oscillations. Recently, the AUSMDV + scheme was developed to improve the numerical stability and accuracy of the AUSMDV scheme. For the AUSMDV + scheme, the AUSMD + and the AUSMV + schemes were introduced as the accurate and the moderately dissipative schemes, respectively. In this paper, the AUSMDV2 + scheme is presented as an improved version of the AUSMDV + scheme. In the AUSMDV2 + scheme, a newly proposed mass flux function is used for both the AUSMD + and the AUSMV + schemes. Additionally, a new normalized weighting function is proposed to control the amount of dissipation. Many tests show that the proposed scheme's stability and accuracy are easily controlled by varying the constant parameter κ between 0.1 and 1.0. Furthermore, the dissipation mechanism of the scheme is studied by employing linearized discrete analysis on the odd–even decoupling problem. The resulting recursive equations show that the proposed scheme can effectively damp out all perturbations over time. Finally, the scheme's numerical stability and accuracy are examined in several test cases on structured and unstructured grids with the first- and second-order accurate methods. The results show that the proposed scheme is stable, accurate, and suitable for solving a wide range of high-speed compressible flow problems. [ABSTRACT FROM AUTHOR]

One- and two-dimensional sine-Gordon equation in non-homogeneous media is considered. Sine-Gordon equation exhibits soliton-like solution whose existence and behaviour in non-homogeneous media is studied. Various non-homogeneous media are discussed which include nonlinear smooth as well as discontinuous density variations. Time-dependent continuous and discontinuous density variations which can model geodynamical processes like lithospheric deformation, fault dynamics, etc. are also discussed. In one dimension, dynamics of kink which are topological soliton is studied under such density variations, whereas in two dimensions, circular-ring soliton dynamics has been scrutinized. The governing sine-Gordon equation is solved numerically using higher order Legendre polynomial-based spectral element method. Spectral stability analysis of the numerical scheme shows the strong dependence of a critical time step not only on density value but also on the nature of the distribution. It is shown that for smooth density variation the critical time step reduces drastically which makes explicit scheme very expensive. To reduce the computational cost, unconditionally stable fully implicit spectral element scheme is proposed which is also energy conservative. [ABSTRACT FROM AUTHOR]

An algorithm for the reaction-diffusion system including model problems Brusselator, Schnakenberg and Gray-Scott is introduced. The integration of the system is managed by combining the Crank-Nicolson method in time and the cubic trigonometric B-spline collocation method in space. Our aim here is to provide a new code to understand and implement reaction-diffusion-type events. Some problems chosen from the literature to illustrate the efficiency of the algorithm are studied for each model problem. [ABSTRACT FROM AUTHOR]

The simple low-dissipation advection upwind splitting method (SLAU) scheme is a parameter-free, low-dissipation upwind scheme that has been applied in a wide range of aerodynamic numerical simulations. In spite of its successful applications, the SLAU scheme could be showing shock instabilities on unstructured grids, as many other contact resolved upwind schemes. Therefore, a hybrid upwind flux scheme is devised for improving the shock stability of SLAU scheme, without compromising on accuracy and low Mach number performance. Numerical flux function of the hybrid scheme is written in a general form, in which only the scalar dissipation term is different from that of the SLAU scheme. The hybrid dissipation term is defined by using a differentiable multidimensional-shock-detection pressure weight function, and the dissipation term of SLAU scheme is combined with that of the Van Leer scheme. Furthermore, the hybrid dissipation term is only applied for the solution of momentum fluxes in numerical flux function. Based on the numerical test results, the hybrid scheme is deemed to be a successful improvement on the shock stability of SLAU scheme, without compromising on the efficiency and accuracy. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Stability of Saul'yev's methods for heat conduction with nonuniform grids is investigated. Though these methods are known to be unconditionally stable with uniform grids, the author shows that their stability with nonuniform grids depends on time step Δτ, space intervals ΔX, and ratios of neighboring space intervals. The author also shows that physical reality is broken when Patankar's positive-coefficients rule is not satisfied, even if the methods are applied to uniform grids. This article presents a stability criterion for Saul'yev's methods with both nonuniform and uniform grids. [ABSTRACT FROM AUTHOR]