Shaoshuai Chu, Olyana A. Kovyrkina, Alexander Kurganov, Vladimir V. Ostapenko
Abstract We study experimental convergence rates of three shock‐capturing schemes for hyperbolic systems of conservation laws: the second‐order central‐upwind (CU) scheme, the third‐order Rusanov‐Burstein‐Mirin (RBM), and the fifth‐order alternative weighted essentially non‐oscillatory (A‐WENO) scheme. We use three imbedded grids to define the experimental pointwise, integral, and convergence rates. We apply the studied schemes to the shallow water equations and conduct their comprehensive numerical convergence study. We verify that while the studied schemes achieve their formal orders of accuracy on smooth solutions, after the shock formation, a part of the computed solutions is affected by shock propagation and both the pointwise and integral convergence rates reduce there. Moreover, while the convergence rates for the CU and A‐WENO schemes, which rely on nonlinear stabilization mechanisms, reduce to the first order, the RBM scheme, which utilizes a linear stabilization, is clearly second‐order accurate. Finally, relying on the conducted experimental convergence rate study, we develop two new combined schemes based on the RBM and either the CU or A‐WENO scheme. The obtained combined schemes can achieve the same high order of accuracy as the RBM scheme in the smooth areas while being non‐oscillatory near the shocks.
{"title":"Experimental convergence rate study for three shock‐capturing schemes and development of highly accurate combined schemes","authors":"Shaoshuai Chu, Olyana A. Kovyrkina, Alexander Kurganov, Vladimir V. Ostapenko","doi":"10.1002/num.23053","DOIUrl":"https://doi.org/10.1002/num.23053","url":null,"abstract":"Abstract We study experimental convergence rates of three shock‐capturing schemes for hyperbolic systems of conservation laws: the second‐order central‐upwind (CU) scheme, the third‐order Rusanov‐Burstein‐Mirin (RBM), and the fifth‐order alternative weighted essentially non‐oscillatory (A‐WENO) scheme. We use three imbedded grids to define the experimental pointwise, integral, and convergence rates. We apply the studied schemes to the shallow water equations and conduct their comprehensive numerical convergence study. We verify that while the studied schemes achieve their formal orders of accuracy on smooth solutions, after the shock formation, a part of the computed solutions is affected by shock propagation and both the pointwise and integral convergence rates reduce there. Moreover, while the convergence rates for the CU and A‐WENO schemes, which rely on nonlinear stabilization mechanisms, reduce to the first order, the RBM scheme, which utilizes a linear stabilization, is clearly second‐order accurate. Finally, relying on the conducted experimental convergence rate study, we develop two new combined schemes based on the RBM and either the CU or A‐WENO scheme. The obtained combined schemes can achieve the same high order of accuracy as the RBM scheme in the smooth areas while being non‐oscillatory near the shocks.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135916102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider two kinds of numerical quadrature formulas of Gauss type and Newton‐Cotes type, which are required in the real computation of Lagrange–Galerkin scheme for the Navier–Stokes problem. The Lagrange–Galerkin scheme with numerical quadrature, which has been used practically but not fully analyzed, is proved to be convergent at least for Gauss type quadrature under a condition on the time increment. As for the scheme with Newton‐Cotes type quadrature, it has more smooth convergent property than that of Gauss type, whose reason is discussed.
{"title":"Behavior of Lagrange‐Galerkin solutions to the Navier‐Stokes problem for small time increment","authors":"M. Tabata, Shinya Uchiumi","doi":"10.1002/num.23051","DOIUrl":"https://doi.org/10.1002/num.23051","url":null,"abstract":"We consider two kinds of numerical quadrature formulas of Gauss type and Newton‐Cotes type, which are required in the real computation of Lagrange–Galerkin scheme for the Navier–Stokes problem. The Lagrange–Galerkin scheme with numerical quadrature, which has been used practically but not fully analyzed, is proved to be convergent at least for Gauss type quadrature under a condition on the time increment. As for the scheme with Newton‐Cotes type quadrature, it has more smooth convergent property than that of Gauss type, whose reason is discussed.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"4295 - 4316"},"PeriodicalIF":3.9,"publicationDate":"2023-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48447654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01Epub Date: 2022-09-23DOI: 10.1177/00471178221122957
Stephane J Baele, Elise Rousseau
This paper offers a multi-dimensional analysis of the ways and extent to which the US president and UK prime minister have securitized the Covid-19 pandemic in their public speeches. This assessment rests on, and illustrates the merits of, both an overdue theoretical consolidation of Securitization Theory's (ST) conceptualization of securitizing language, and a new methodological blueprint for the study of 'securitizing semantic repertoire'. Comparing and contrasting the two leaders' respective securitizing semantic repertoires adopted in the early months of the coronavirus outbreak shows that securitizing language, while very limited, has been more intense in the UK, whose repertoire was structured by a biopolitical imperative to 'save lives' in contrast to the US repertoire centred on the 'war' metaphor.
{"title":"At war or saving lives? On the securitizing semantic repertoires of Covid-19.","authors":"Stephane J Baele, Elise Rousseau","doi":"10.1177/00471178221122957","DOIUrl":"10.1177/00471178221122957","url":null,"abstract":"<p><p>This paper offers a multi-dimensional analysis of the ways and extent to which the US president and UK prime minister have securitized the Covid-19 pandemic in their public speeches. This assessment rests on, and illustrates the merits of, both an overdue theoretical consolidation of Securitization Theory's (ST) conceptualization of securitizing language, and a new methodological blueprint for the study of 'securitizing semantic repertoire'. Comparing and contrasting the two leaders' respective securitizing semantic repertoires adopted in the early months of the coronavirus outbreak shows that securitizing language, while very limited, has been more intense in the UK, whose repertoire was structured by a biopolitical imperative to 'save lives' in contrast to the US repertoire centred on the 'war' metaphor.</p>","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"34 1","pages":"201-227"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9510966/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91026446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we present a finite volume scheme preserving invariant‐region‐property (IRP) for a class of semilinear parabolic equations with anisotropic diffusion coefficient on distorted meshes. The diffusion term is discretized by the finite volume scheme preserving the discrete maximum principle, and the time derivative is discretized by the backward Euler scheme. For the nonlinear system, a specially designed iteration is proposed to preserve the IRP. The IRPs are proved for both, the finite volume scheme and the nonlinear iteration. Numerical examples are presented to verify the accuracy and IRP of our scheme.
{"title":"A finite volume scheme preserving the invariant region property for a class of semilinear parabolic equations on distorted meshes","authors":"Huifang Zhou, Yuanyuan Liu, Z. Sheng","doi":"10.1002/num.23050","DOIUrl":"https://doi.org/10.1002/num.23050","url":null,"abstract":"In this article, we present a finite volume scheme preserving invariant‐region‐property (IRP) for a class of semilinear parabolic equations with anisotropic diffusion coefficient on distorted meshes. The diffusion term is discretized by the finite volume scheme preserving the discrete maximum principle, and the time derivative is discretized by the backward Euler scheme. For the nonlinear system, a specially designed iteration is proposed to preserve the IRP. The IRPs are proved for both, the finite volume scheme and the nonlinear iteration. Numerical examples are presented to verify the accuracy and IRP of our scheme.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"4270 - 4294"},"PeriodicalIF":3.9,"publicationDate":"2023-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46724969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article is concerned with numerical study of a coupled system of Gross–Pitaevskii equations which describes the spin‐orbit‐coupled Bose–Einstein condensates. Due to the fact that this system possesses the total mass and energy conservation property and often appears in high dimensions, it brings a significant burden in designing and analyzing a suitable numerical scheme for solving the coupled Gross–Pitaevskii equations (CGPEs). In this article, an implicit finite difference scheme is proposed to solve the CGPEs, which is proved to be uniquely solvable, mass‐ and energy‐conservative in the discrete sense. In particular, it is proved in a rigorous way that, without any grid‐ratio restriction, the scheme is stable and convergent at the rate of O(h2+τ2)$$ Oleft({h}^2+{tau}^2right) $$ with time step τ$$ tau $$ and mesh size h$$ h $$ in the maximum norm, while previous works often require certain restriction on the grid ratio and only give the error estimates in the discrete L2$$ {L}^2 $$ norm or H1$$ {H}^1 $$ norm which could not imply the maximum error estimate. Numerical results are carried out to underline the error estimate and conservation laws, and investigate several dynamics of the CGPEs.
{"title":"A mass‐ and energy‐preserving numerical scheme for solving coupled Gross–Pitaevskii equations in high dimensions","authors":"Jianfeng Liu, Q. Tang, Ting-chun Wang","doi":"10.1002/num.23042","DOIUrl":"https://doi.org/10.1002/num.23042","url":null,"abstract":"This article is concerned with numerical study of a coupled system of Gross–Pitaevskii equations which describes the spin‐orbit‐coupled Bose–Einstein condensates. Due to the fact that this system possesses the total mass and energy conservation property and often appears in high dimensions, it brings a significant burden in designing and analyzing a suitable numerical scheme for solving the coupled Gross–Pitaevskii equations (CGPEs). In this article, an implicit finite difference scheme is proposed to solve the CGPEs, which is proved to be uniquely solvable, mass‐ and energy‐conservative in the discrete sense. In particular, it is proved in a rigorous way that, without any grid‐ratio restriction, the scheme is stable and convergent at the rate of O(h2+τ2)$$ Oleft({h}^2+{tau}^2right) $$ with time step τ$$ tau $$ and mesh size h$$ h $$ in the maximum norm, while previous works often require certain restriction on the grid ratio and only give the error estimates in the discrete L2$$ {L}^2 $$ norm or H1$$ {H}^1 $$ norm which could not imply the maximum error estimate. Numerical results are carried out to underline the error estimate and conservation laws, and investigate several dynamics of the CGPEs.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"4248 - 4269"},"PeriodicalIF":3.9,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46907877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a two‐level stabilized finite volume method is developed and analyzed for the steady incompressible magnetohydrodynamic (MHD) equations. The linear polynomial space is used to approximate the velocity, pressure and magnetic fields, and two local Gauss integrations are introduced to overcome the restriction of discrete inf‐sup condition. Firstly, the existence and uniqueness of the solution of the discrete problem in the stabilized finite volume method are proved by using the Brouwer's fixed point theorem. H2$$ {H}^2 $$ ‐stability results of numerical solutions are also presented. Secondly, optimal error estimates of numerical solutions in H1$$ {H}^1 $$ and L2$$ {L}^2 $$ ‐norms are established by using the energy method and constructing the corresponding dual problem. Thirdly, the stability and convergence of two‐level stabilized finite volume method for the stationary incompressible MHD equations are provided. Theoretical findings show that the two‐level method has the same accuracy as the one‐level method with the mesh sizes h=𝒪(H2) . Finally, some numerical results are provided to identify with the established theoretical findings and show the performances of the considered numerical schemes.
{"title":"Two‐level stabilized finite volume method for the stationary incompressible magnetohydrodynamic equations","authors":"X. Chu, Chuanjun Chen, T. Zhang","doi":"10.1002/num.23043","DOIUrl":"https://doi.org/10.1002/num.23043","url":null,"abstract":"In this paper, a two‐level stabilized finite volume method is developed and analyzed for the steady incompressible magnetohydrodynamic (MHD) equations. The linear polynomial space is used to approximate the velocity, pressure and magnetic fields, and two local Gauss integrations are introduced to overcome the restriction of discrete inf‐sup condition. Firstly, the existence and uniqueness of the solution of the discrete problem in the stabilized finite volume method are proved by using the Brouwer's fixed point theorem. H2$$ {H}^2 $$ ‐stability results of numerical solutions are also presented. Secondly, optimal error estimates of numerical solutions in H1$$ {H}^1 $$ and L2$$ {L}^2 $$ ‐norms are established by using the energy method and constructing the corresponding dual problem. Thirdly, the stability and convergence of two‐level stabilized finite volume method for the stationary incompressible MHD equations are provided. Theoretical findings show that the two‐level method has the same accuracy as the one‐level method with the mesh sizes h=𝒪(H2) . Finally, some numerical results are provided to identify with the established theoretical findings and show the performances of the considered numerical schemes.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"4196 - 4220"},"PeriodicalIF":3.9,"publicationDate":"2023-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44329352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper deals with the study of a higher‐order numerical approximation for a class of singularly perturbed convection‐diffusion problems with time delay. The method combines a higher‐Order Difference with an Identity Expansion (HODIE) scheme over a piece‐wise uniform mesh in the spatial direction and the backward Euler method on a uniform mesh for discretization in the temporal direction. A priori bounds for the continuous solution and its derivatives are derived by splitting the solution into regular and singular components. These bounds are useful in the error analysis of the proposed scheme. The present scheme converges ε$$ varepsilon $$ ‐uniformly with the order of convergence one in time and almost second‐order in space direction. Further, to increase the rate of convergence in the time variable, we implemented the Richardson extrapolation technique. Thus, finally, the resultant scheme with Richardson extrapolation in time becomes almost second‐order ε$$ varepsilon $$ ‐uniformly convergent in both the space and time variable. The detailed stability and convergence analysis have been done using the derived a priori estimates. We consider three test problems to validate the predicted theory and show that numerical results are in good agreement with our theoretical findings.
{"title":"Parameter robust higher‐order finite difference method for convection‐diffusion problem with time delay","authors":"Sanjaya Sahoo, Vikas Gupta","doi":"10.1002/num.23039","DOIUrl":"https://doi.org/10.1002/num.23039","url":null,"abstract":"This paper deals with the study of a higher‐order numerical approximation for a class of singularly perturbed convection‐diffusion problems with time delay. The method combines a higher‐Order Difference with an Identity Expansion (HODIE) scheme over a piece‐wise uniform mesh in the spatial direction and the backward Euler method on a uniform mesh for discretization in the temporal direction. A priori bounds for the continuous solution and its derivatives are derived by splitting the solution into regular and singular components. These bounds are useful in the error analysis of the proposed scheme. The present scheme converges ε$$ varepsilon $$ ‐uniformly with the order of convergence one in time and almost second‐order in space direction. Further, to increase the rate of convergence in the time variable, we implemented the Richardson extrapolation technique. Thus, finally, the resultant scheme with Richardson extrapolation in time becomes almost second‐order ε$$ varepsilon $$ ‐uniformly convergent in both the space and time variable. The detailed stability and convergence analysis have been done using the derived a priori estimates. We consider three test problems to validate the predicted theory and show that numerical results are in good agreement with our theoretical findings.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"4145 - 4173"},"PeriodicalIF":3.9,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44388556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose and analyze a second order accurate in time, energy stable numerical scheme for the strongly anisotropic Cahn–Hilliard system, in which a biharmonic regularization has to be introduced to make the equation well‐posed. A convexity analysis on the anisotropic interfacial energy is necessary to overcome an essential difficulty associated with its highly nonlinear and singular nature. The second order backward differentiation formula temporal approximation is applied, combined with Fourier pseudo‐spectral spatial discretization. The nonlinear surface energy part is updated by an explicit extrapolation formula. Meanwhile, the energy stability analysis is enforced by the fact that all the second order functional derivatives of the energy stay uniformly bounded by a global constant. A Douglas‐Dupont type regularization is added to stabilize the numerical scheme, and a careful estimate ensures a modified energy stability with a uniform constraint for the regularization parameter A$$ A $$ . In turn, the combination with an appropriate treatment for the nonlinear double well potential terms leads to a weakly nonlinear scheme. More importantly, such an energy stability is in terms of the interfacial energy with respect to the original phase variable, which enables us to derive an optimal rate convergence analysis.
{"title":"High order accurate and convergent numerical scheme for the strongly anisotropic Cahn–Hilliard model","authors":"Kelong Cheng, Cheng Wang, S. Wise","doi":"10.1002/num.23034","DOIUrl":"https://doi.org/10.1002/num.23034","url":null,"abstract":"We propose and analyze a second order accurate in time, energy stable numerical scheme for the strongly anisotropic Cahn–Hilliard system, in which a biharmonic regularization has to be introduced to make the equation well‐posed. A convexity analysis on the anisotropic interfacial energy is necessary to overcome an essential difficulty associated with its highly nonlinear and singular nature. The second order backward differentiation formula temporal approximation is applied, combined with Fourier pseudo‐spectral spatial discretization. The nonlinear surface energy part is updated by an explicit extrapolation formula. Meanwhile, the energy stability analysis is enforced by the fact that all the second order functional derivatives of the energy stay uniformly bounded by a global constant. A Douglas‐Dupont type regularization is added to stabilize the numerical scheme, and a careful estimate ensures a modified energy stability with a uniform constraint for the regularization parameter A$$ A $$ . In turn, the combination with an appropriate treatment for the nonlinear double well potential terms leads to a weakly nonlinear scheme. More importantly, such an energy stability is in terms of the interfacial energy with respect to the original phase variable, which enables us to derive an optimal rate convergence analysis.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"4007 - 4029"},"PeriodicalIF":3.9,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45157133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Any robust computational scheme for compressible flows must retain the hyperbolicity property or the real‐valued sound speed. Failure to maintain hyperbolicity, or the positivity of the square of the speed of sound, causes nonphysical distortions and the blow‐up of numerical simulations. Strong shock waves and interfacial discontinuities are ubiquitous features of the two‐fluid compressible dynamics that can potentially induce positivity‐related failure in a simulation. This article presents a positivity‐preserving algorithm for a high‐order, primitive variable‐based, weighted essentially non‐oscillatory finite difference scheme. The positivity preservation relies on a flux limiting technique that locally adapts high‐order fluxes towards the first order to retain the physical bounds of the solution without loss of high‐order convergence. This positivity preserving scheme has been devised and implemented up to eleventh order in one and two dimensions for a two‐fluid compressible model that consists of a single mass, momentum, and energy equations, as well as an advection of material parameters for capturing the interfaces. Several one‐ and two‐dimensional challenging test problems verify the performance. The scheme effectively retains high order accuracy while allowing for the simulation of several challenging problems that otherwise could not be successfully solved using the base scheme, without any penalty on the CFL condition requirement, and without any significant impact on the CPU times. The scheme represents the first high‐order (up to 11‐order) hyperbolicity‐preserving scheme for the considered two‐fluid compressible flows in the fully Eulerian formulation. The inherent efficiency of finite differences and the new robust positivity preserving quality enable modeling other challenging problems of two‐fluid and two‐phase problems.
{"title":"A positivity preserving high‐order finite difference method for compressible two‐fluid flows","authors":"Daniel Boe, Khosro Shahbazi","doi":"10.1002/num.23037","DOIUrl":"https://doi.org/10.1002/num.23037","url":null,"abstract":"Any robust computational scheme for compressible flows must retain the hyperbolicity property or the real‐valued sound speed. Failure to maintain hyperbolicity, or the positivity of the square of the speed of sound, causes nonphysical distortions and the blow‐up of numerical simulations. Strong shock waves and interfacial discontinuities are ubiquitous features of the two‐fluid compressible dynamics that can potentially induce positivity‐related failure in a simulation. This article presents a positivity‐preserving algorithm for a high‐order, primitive variable‐based, weighted essentially non‐oscillatory finite difference scheme. The positivity preservation relies on a flux limiting technique that locally adapts high‐order fluxes towards the first order to retain the physical bounds of the solution without loss of high‐order convergence. This positivity preserving scheme has been devised and implemented up to eleventh order in one and two dimensions for a two‐fluid compressible model that consists of a single mass, momentum, and energy equations, as well as an advection of material parameters for capturing the interfaces. Several one‐ and two‐dimensional challenging test problems verify the performance. The scheme effectively retains high order accuracy while allowing for the simulation of several challenging problems that otherwise could not be successfully solved using the base scheme, without any penalty on the CFL condition requirement, and without any significant impact on the CPU times. The scheme represents the first high‐order (up to 11‐order) hyperbolicity‐preserving scheme for the considered two‐fluid compressible flows in the fully Eulerian formulation. The inherent efficiency of finite differences and the new robust positivity preserving quality enable modeling other challenging problems of two‐fluid and two‐phase problems.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"39 1","pages":"4087 - 4125"},"PeriodicalIF":3.9,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46168113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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