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The numerical shock instability problem significantly limits the application of the finite volume method in hypersonic flow simulations, especially for high-order schemes, which will encounter more severe shock instability problems. In the previous work, the MSAT (matrix stability analysis tool) was introduced to quantitatively analyze the numerical shock instability problem. However, the utilization of the three-point reconstruction stencil in the original version of MSAT presents challenges in conducting investigations into high-order schemes. The current work provides an extension of MSAT by extending the reconstruction stencil to five points, a challenging but necessary endeavor for high-order schemes. This extension enables MSAT to provide analyses for schemes with high-order reconstruction methods. Furthermore, the updated MSAT can be utilized to investigate the impact of characteristic variable reconstruction on the shock instability problem. Results demonstrate the reliability of the updated MSAT and reveal novel characteristics in the shock instability of high-order schemes. Consequently, the updated MSAT will contribute to the investigation of shock instabilities for high-order schemes, ultimately aiding in mitigating this challenging problem.

The simulation of supersonic or hypersonic flows often suffers from numerical shock instabilities if the flow field contains strong shocks, limiting the further application of shock-capturing schemes. In this paper, we develop the unified matrix stability analysis method for schemes with three-point stencils and present MSAT (matrix stability analysis tool), an open-source tool to quantitatively analyze the shock instability problem. Based on the finite-volume approach on the structured grid, MSAT can be employed to investigate the mechanism of the shock instability problem, evaluate the robustness of numerical schemes, and then help to develop robust schemes. Also, MSAT has the ability to analyze the practical simulation of supersonic or hypersonic flows, evaluate whether it will suffer from shock instabilities, and then assist in selecting appropriate numerical schemes accordingly. As a result, MSAT is a helpful tool that can investigate the shock instability problem and help to cure it.

In present paper, a new technique is developed named Uniform Algebraic Trigonometric (UAT) tension B-spline based DQM, for the better numerical approximation of coupled 1D and coupled 2D Burgers’ equation. The formula of UAT tension B-spline of order 4 is developed in this paper, by using above-mentioned basis function. Strong stability preserving SSP-RK 43 method is implemented to solve obtained ordinary differential equations. To check the effectiveness and accuracy of the proposed scheme, L2 and L∞error norms are used. Four numerical experiments are discussed to put light upon applicability of scheme. By means of tables and figures, results obtained are given. Obtained results are in a good agreement with existing ones. Stability of proposed scheme is also checked by implementing the Matrix stability analysis method. This novel scheme will surely open some new dimensions in the field of numerical approximation, for other researchers in their future work.

This paper proposes a new scheme termed as modified extended cubic B-Spline differential quadrature (mECDQ) method for time dependent partial differential equations. Specially, the numerical computation of the Burgers’ equation is obtained using mECDQ method. First of all the modified extended cubic B-splines are used as a set of basis function in DQ to evaluate the weighting coefficients. The mECDQ method converts the initial boundary value system of Burgers’ equation into a initial value system of ordinary differential equations (ODEs). The resulting system is solved by using an optimal five stage four order strong stability preserving Runge–Kutta method (SSP-RK54). The accuracy and efficiency of the method is illustrated by considering five test problems. The proposed results are compared with the exact solutions in terms of L2 and L∞ error norms and the existing results. The mECDQ scheme produces better results than the results due to almost all the existing schemes. The stability analysis of the scheme is also carried out using the matrix stability analysis method for various grid points. This shows that mECDQ scheme is conditionally stable.

The positive-definite stability analysis (PDSA) is presented as a technique complementary to the companion-matrix stability analysis (CMSA). The PDSA is used to analyze the stability of marching-on-in-time (MOT) schemes. The heart of the PDSA is formed by the analysis on particular linear combinations of interaction matrices from an MOT scheme, which are assumed to be real-valued. If these are all positive definite, then the PDSA guarantees the stability of the scheme. The PDSA can be of a lower complexity than the full CMSA. The construction of the PDSA is shown and applied to two numerical examples. [ABSTRACT FROM AUTHOR]

The aim of this study is to develop compact difference method to approximate parabolic PDEs of fourth order equipped with Dirichlet and Neumann boundary conditions involving half-step points. The proposed method converges quaternary and quadratically in space and time, respectively. The imbedding technique has been applied to approximate derivative terms of lower order by means of the governing differential equation to deduce the high-order method. The primary utility of this new discretization is that it can be straightaway applied to problems with singularities without necessitating fictitious nodes or special approach which has consequently lowered computational complicacy. We have examined linear stability of the proposed three-level implicit difference scheme using matrix stability analysis. In addition, we also obtained the solution of the first-order spatial derivative which is of significance in several physical problems. The efficacy of the proposed approximation is confirmed through numerical tests performed on a collection of physically relevant problems comprising the Euler Bernoulli beam equation and the highly nonlinear good Boussinesq equation. Numerical experiments evidently exhibit that the method provides more accurate results in contrast with the existing numerical techniques. The present method is able to simulate well the complex and intriguing long time dynamics of the good Boussinesq equation. [ABSTRACT FROM AUTHOR]

The contrast current density volume integral equation, discretized with piecewise constant spatial basis and test functions and Dirac-delta temporal test functions and the piecewise polynomial temporal basis functions, results in a causal implicit marching-on-intime scheme that we refer to as the marching-on-in-time contrast current density volume integral equation (MOT-JVIE). The companion matrix stability analysis of the MOT-JVIE solver shows that for a fixed spatial and temporal step size, the stability is independent of the scatterer's dielectric contrast for quadratic spline temporal basis functions. Whereas, Lagrange and cubic spline exhibit instabilities at higher contrast. We relate this stability performance to the expansion and testing procedure in time. We further illustrate the capabilities of the MOT-JVIE based on quadratic spline temporal basis functions by: comparing the MOT-JVIE solution to time-domain results from literature and frequency-domain results from a commercial combined field integral equation solver. Finally, we present a long time sequence for a high-contrast scatterer discretized with 24,000 spatial unknowns. [ABSTRACT FROM AUTHOR]

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]

In December 2019, the newest infection, named novel coronavirus COVID-19, appeared, and it became a challenge to the world. In this paper, we investigated a mathematical model that explains the transmission of COVID- 19 with the help of ordinary differential equations. Here, we divided the entire model into eight different compartments, namely SEI1I2I3QHR model based on three different age groups. Equilibrium analysis with positivity and boundedness are also investigated. The reproduction number is analyzed using Vandedressche's next-generation matrix. Stability analysis for disease-free equilibrium (DFE) is studied. We have taken some basic parameters with their primary conditions from the Ethiopian Ministry of Health Institute from 29 February 2021 to 07 June 2021. [ABSTRACT FROM AUTHOR]

This study presents a numerical solution of Fisher's equation. For time integration, Crank–Nicolson and fourth-order one-step implicit schemes are used and for space discretization, quintic B-spline Collocation and quintic B-spline Galerkin methods are employed. The truncation error is analyzed and the stability of the suggested methods is discussed matrix stability analysis. Three examples are studied to compare the present results with existing numerical results by computing error norm L ∞ and the order of convergence. The obtained results show that the proposed methods are satisfactorily efficient in terms of accuracy. [ABSTRACT FROM AUTHOR]

In the present paper, a Barycentric Lagrange interpolation polynomial-based Differential quadrature method is applied to get the numerical solution of non-linear coupled 2D But-gers' equation, For this process, first of all, the coupled Burgers' equation is dealt with generalized formula, By using the obtained generalized formula, required weighting coefficients in the differential quadrature method are easily calculated, Afterward, the reduced set of ordinary differential equations is solved by using the SSP-RK43 scheme, To test the efficiency of the newly applied method, the L., error norm has been computed, To test the stability of the proposed method, the Matrix stability analysis method is implemented, which shows that the developed method is unconditionally stable. [ABSTRACT FROM AUTHOR]

Present paper deals with the numerical solution of coupled 1D Burgers' equation by implementing the Non-Uniform Algebraic Hyperbolic (NUAH) B-spline Differential Quadrature Method. In the present paper, the spatial variable discretization is done using NUAH B-spline, and the obtained system of ODE is dealt with using the SSP-RK43 scheme. To get the improvised results, the concept of modified cubic NUAH B-spline is incorporated. To test the effectiveness and accuracy of the proposed scheme, numerical examples are discussed. Stability analysis of the proposed scheme is investigated by the matrix stability analysis method. The present regime is worthwhile to deal with some complex natured PDEs, where finding the analytical solution is cumbersome. [ABSTRACT FROM AUTHOR]

A simple HLLE-type scheme is proposed for all Mach number flows. In the proposed scheme, no extra wave structure is added in the HLLE scheme to resolve the shear wave while the contact wave is resolved by adding a wave structure similar to the HLLEM scheme. The resolution of the shear layers and the flow features at low Mach numbers is achieved by a velocity reconstruction method based on the face normal Mach number. Robustness against the numerical instabilities is achieved by scaling the velocity reconstruction function in the vicinity of shock with a multi-dimensional pressure sensor. The ability of the proposed scheme to resolve low Mach flow features is demonstrated through asymptotic analysis while the stability of the proposed scheme for strong shock is demonstrated through linear perturbation and matrix stability analyses. A set of numerical test cases are solved to show that the proposed scheme is free from numerical shock instability problems at high speeds and is capable of resolving the flow features at very low Mach numbers. • A new velocity reconstruction method for an all Mach number HLLE scheme. • Velocity reconstruction based on face normal Mach number and pressure function. • Velocity reconstruction resolves shear layers and low Mach flow features while presrving shock stability. • HLLEM type method for resolving contact wave while preserving shock stability. • Analytical and numerical demonstration of all Mach capability of the proposed scheme. [ABSTRACT FROM AUTHOR]

We propose a differential quadrature method (DQM) based on cubic hyperbolic B-spline basis functions for computing 3D wave equations. This method converts the problem into a system of ODEs. We use an optimum five-stage and order four SSP Runge-Kutta (SSPRK-(5,4)) scheme to solve the obtained system of ODEs. The matrix stability analysis is also investigated. The accuracy and efficiency of the proposed method are demonstrated via three numerical examples. It has been found that the proposed method gives more accurate results than the existing methods. The main purpose of this work is to present an accurate, economically easy-to-implement, and stable technique for solving hyperbolic partial differential equations. [ABSTRACT FROM AUTHOR]

In this work, a numerical regime based upon the Non-Uniform Algebraic Hyperbolic Bspline is developed with the aid of the differential quadrature method named NUAH B-spline DQM to fetch the numerical approximation of non-linear coupled ZD Burgers equation. In the first step, the spatial discretization is mentioned using the NUAH B-spline of order 4: for improvised results, modified formulae are implemented, In the second step, the reduced system of ordinary differential equations is dealt using SSP-RK43 regime. The proposed regime is analyzed for stability also. Finally, the effectiveness and adaptability of the proposed regime are confirmed with the aid of numerical experiments, and fetched results are compared with the existing outcomes in the literature. It is observed that the present scheme is unconditionally stable by implementing the concept of the matrix stability analysis method. It is found that the present regime is easy to implement and produces accurate results. [ABSTRACT FROM AUTHOR]

The aim of present study is to develop a numerical scheme by using the notion of Barycentric Lagrange interpolation based Differential quadrature method to solve the coupled 1D Burgers' equation. This method reduced the mentioned partial differential equation into the set of ordinary differential equations, which can be dealt by the SSP-RK43 scheme. The proposed method has been implemented upon the different numerical examples in order to test the accuracy and effectiveness of the proposed method. It is observed that obtained results are in good compatibility with the exact solution and are better than the previous results. The stability of the proposed method is also discussed by using the matrix stability analysis method, which represents that the proposed method is unconditionally stable. [ABSTRACT FROM AUTHOR]

In this, research work. we present a novel technique named Hyperbolic B-spline based Differential quadrature method to get the numerical approximations of the 1 D and 2D coupled nonlinear Schrodinger equations. The coupled non-linear Schrodinger equation represents the range of physical phenomena with the aid of a model equation for the optical fiber with the linear birefringence. In present paper. Differential quadrature method is applied over the mentioned PDEs and reduced set of ordinary differential equations is solved by SSP-RK43 scheme. Accuracy and effectiveness of the proposed scheme is checked by L∞ error norm and conservation of mass properly is also tested. In present paper. elastic property is also checked. whether the wave amplitudes have restored their shapes or not for the different used parameters and stability of the proposed scheme is checked via matrix stability analysis method. which indicates that the present method has the property of unconditionally stable. [ABSTRACT FROM AUTHOR]

Present paper deals with the numerical solution of coupled 2D non-linear Burgers equation by implementing the notion of modified cubic Hyperbolic B-spline based Differential quadrature method. Feasibility and accuracy of proposed scheme is checked by using L2 and L∞ error norms. Comparison of errors is shown for the mentioned examples. Reduced set of ordinary differential equations is tackled by using SSP-RK43 scheme. Stability of the proposed scheme is checked by using the Matrix stability analysis method. It is noticed that proposed numerical scheme produced the results with accuracy and it is quite easy to implement. [ABSTRACT FROM AUTHOR]

The paper devises a new robust carbuncle-free Roe Riemann solver for strong shock, different from hybrid method, entropy fix, Liou’s conjecture (J Comput Phys 160:623-648, 2000) and artificial viscosity (Rodionov in J Comput Phys 345:308-329, 2017). Roe scheme encounters carbuncle phenomenon, violates entropy condition and lacks positivity property. The remedy integrates shear viscosity into momentum flux to damp undesirable perturbation and prevents unstable vorticity mode from triggering shock instability. Compared to HLL scheme, shear viscosity is properly established through dimensional analysis and analogy method. The non-linear wave speeds are slightly modified by incorporating the neighboring cell information for positive conservation. The pressure-based sensing function is applied to preserve shear layer while keeping shock robustness. The resulting scheme is very easily implemented by converting the existing Roe code. The matrix stability analysis confirms that this approach is shock-stable and more robust than entropy fix. A series of numerical results demonstrate its potential features: positivity-preserving property, entropy-satisfying property, accurate boundary-layer resolution and high robustness against shock instability. In addition, it signifies that the reason to cause shock instability for Roe scheme may be not the pressure difference term in mass flux, but the inadequate shear viscosity. [ABSTRACT FROM AUTHOR]

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This article concerns with the numerical investigations of the reaction–diffusion systems (RDSs) arising in the study of pattern formation in biological and chemical systems with the employment of the quartic-trigonometric B-spline functions. The computationally numerical scheme uses collocation method which is established by a relatively new B-splines for the spatial discretizations and, for time integration Crank–Nicolson technique is adapted. Therefore, solutions of the RDSs are assembled by the wholly discretized space-time scheme. A matrix stability analysis is performed for the numerical scheme after linearization process. Experimental cases include Brusselator model, Gray–Scott model, Schnakenberg model as well as a linear problem in one-dimensional domain. Numerical solutions are compared to the existing studies. Spatial pattern formation is demonstrated by present computational algorithm. [ABSTRACT FROM AUTHOR]

A multiple locally 1-D (MLOD) finite-difference time-domain (FDTD) method for inhomogeneous coupled transmission lines and stability analyses are presented. The method is aptly called the MLOD coupled line (CL)-FDTD method. Various split matrices are proposed, and the corresponding update equations are formulated and discussed. All the proposed split matrices yield implicit electric field update equations with tridiagonal or block tridiagonal matrices on the left-hand sides. For more efficiency, the block tridiagonal matrices for implicit electric field may be reformulated and replaced with tridiagonal matrices for implicit magnetic field. The stability analysis is first performed using the von Neumann method in the Fourier domain. It is shown that the von Neumann method alone may not be sufficient to ascertain stability for inhomogeneous media. To include media inhomogeneity, the two-media reduced-matrix stability analysis is proposed. It allows us to efficiently analyze the key stability characteristics in inhomogeneous media and is useful for quick detection of any potential instability that is not apparent via the von Neumann method. The stability characteristics with variation of media parameters are also investigated and discussed. The numerical results are provided to validate the stability and accuracy of the proposed MLOD CL-FDTD method with various split matrices. [ABSTRACT FROM AUTHOR]

This chapter is concerned with the problem of the functional observerbased sliding mode control (SMC) design for parametric uncertain discrete-time delayed stochastic systems including mismatched parameter uncertainty in the state matrix and in the delayed state matrix. Stability analysis of the sliding function is presented in a time delayed stochastic system with the linear matrix inequality (LMI) approach. Moreover, it is shown that the state trajectories can be driven onto the specified sliding surface despite the presence of state delay, unmatched parameter uncertainty, and stochastic noise in the system. The research is motivated by the fact that system states are not always accessible for the state feedback. Therefore, SMC is estimated using the functional observer technique. To mitigate the side effect of parametric uncertainty on the estimation error, a sufficient condition of stability is proposed based on Gershgorin's circle theorem. The claims made are validated through numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, an attempt is made to understand the transmission dynamics of Ebola virus disease (EVD) incorporating fuzziness in all biological parameters due to its natural variability. To characterize the transmission trajectories of Ebola outbreak, we propose and analyze two SEIR and SEIRHD type transmission models. Using triangular fuzzy numbers for the imprecise parameters, we first study the existence of the equilibria and their stability. Both of the model have two equilibria, namely the disease-free and endemic. Stability of the disease-free and endemic equilibria is related with basic reproduction number that has been calculated from next generation matrix. Stability analysis of the system shows that the disease free equilibrium is locally as well as globally asymptotically stable when the basic reproduction number is less than unity. Under some additional conditions, the model system becomes locally asymptotically stable at unique endemic equilibrium when basic reproduction number is greater than unity. Finally, we perform some numerical experiments to justify the theoretical estimate. [ABSTRACT FROM AUTHOR]

This study is concerned with the problem of the functional observer‐based sliding mode control (SMC) design for parametric uncertain discrete‐time delayed stochastic systems includes mismatched parameter uncertainty in the state matrix and in the delayed state matrix. Stability analysis of sliding function is presented in the time delayed stochastic system with a linear matrix inequality approach. Moreover, it is shown that the state trajectories can be driven onto the specified sliding surface despite the presence of state delay, unmatched parameter uncertainty and stochastic noise in the system. The research is motivated by the fact that the system states are not always accessible for the state feedback. Therefore, SMC is estimated using the functional observer technique. To mitigate the side effect of the parameter uncertainty on the estimation error, a sufficient condition of stability is proposed based on Gershgorin disc theorem. The claims made are validated through numerical simulations. [ABSTRACT FROM AUTHOR]