We prove a moderate deviation principle for stochastic differential equations (SDEs) with non-Lipschitz conditions. As an application of our result, we also study the stochastic Hamiltonian systems.
{"title":"A moderate deviation principle for stochastic Hamiltonian systems","authors":"Jie Xu, Jiayin Gong, Jie Ren","doi":"10.1051/ps/2023009","DOIUrl":"https://doi.org/10.1051/ps/2023009","url":null,"abstract":"We prove a moderate deviation principle for stochastic differential equations (SDEs) with non-Lipschitz conditions. As an application of our result, we also study the stochastic Hamiltonian systems.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"84 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79808973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The random mapping construction of strong stationary times is applied here to finite Heisenberg random walks over $ZZ_M$, for odd $Mgeq 3$. When they correspond to $3times 3$ matrices, the strong stationary times are of order $M^6$, estimate which can be improved to $M^4$ if we are only interested in the convergence to equilibrium of the last column. Simulations by Chhaïbi suggest that the proposed strong stationary time is of the right $M^2$ order. These results are extended to $Ntimes N$ matrices, with $Ngeq 3$. All the obtained bounds are thought to be non-optimal, nevertheless this original approach is promising, as it relates the investigation of the previously elusive strong stationary times of such random walks to new absorbing Markov chains with a statistical physics flavor and whose quantitative study is to be pushed further. In addition, for $N=3$, a strong equilibrium time is proposed in the same spirit for the non-Markovian coordinate in the upper right corner. This result would extend to separation discrepancy the corresponding fast convergence for this coordinate in total variation and open a new method for the investigation of this phenomenon in higher dimension.
{"title":"Strong stationary times for finite Heisenberg walk","authors":"L. Miclo","doi":"10.1051/ps/2023008","DOIUrl":"https://doi.org/10.1051/ps/2023008","url":null,"abstract":"The random mapping construction of strong stationary times is applied here to finite Heisenberg random walks over $ZZ_M$, for odd $Mgeq 3$.\u0000When they correspond to $3times 3$ matrices, the strong stationary times are of order $M^6$, estimate which can be improved to $M^4$\u0000if we are only interested in the convergence to equilibrium of the last column.\u0000Simulations by Chhaïbi suggest that the proposed strong stationary time is of the right $M^2$ order.\u0000These results are extended to $Ntimes N$ matrices, with $Ngeq 3$.\u0000All the obtained bounds are thought to be non-optimal, nevertheless this original approach is promising, as it relates the investigation of the previously elusive strong stationary times\u0000of such random walks to new absorbing Markov chains with a statistical physics flavor and whose quantitative study is to be pushed further.\u0000In addition, for $N=3$, a strong equilibrium time is proposed in the same spirit for the non-Markovian coordinate in the upper right corner.\u0000This result would extend to separation discrepancy the corresponding fast convergence for this coordinate in total variation\u0000and open a new method for the investigation of this phenomenon in higher dimension.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"56 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80209737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Elastic flow for closed curves can involve significant deformations. Mesh-based approximation schemes require tangentially redistributing vertices for long-time computations. We present and analyze a method that uses the Dirichlet energy for this purpose. The approach effectively also penalizes the length of the curve, and equilibrium shapes are equivalent to stationary points of the elastic energy augmented with the length functional. Our numerical method is based on linear parametric finite elements. Following the lines of Deckelnick and Dziuk [ Math. Comp. 78 (2009) 645–671] we prove convergence and establish error estimates, noting that the addition of the Dirichlet energy simplifies the analysis in comparison with the length functional. We also present a simple semi-implicit time discretization and discuss some numerical results that support the theory.
{"title":"Convergence of a scheme for an elastic flow with tangential mesh movement","authors":"Paola Pozzi, Bjoern Stinner","doi":"10.1051/m2an/2022091","DOIUrl":"https://doi.org/10.1051/m2an/2022091","url":null,"abstract":"Elastic flow for closed curves can involve significant deformations. Mesh-based approximation schemes require tangentially redistributing vertices for long-time computations. We present and analyze a method that uses the Dirichlet energy for this purpose. The approach effectively also penalizes the length of the curve, and equilibrium shapes are equivalent to stationary points of the elastic energy augmented with the length functional. Our numerical method is based on linear parametric finite elements. Following the lines of Deckelnick and Dziuk [ Math. Comp. 78 (2009) 645–671] we prove convergence and establish error estimates, noting that the addition of the Dirichlet energy simplifies the analysis in comparison with the length functional. We also present a simple semi-implicit time discretization and discuss some numerical results that support the theory.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"688 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135742653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a priori error estimates in the L 2 norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the L 2 norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the L 2 norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in L 2 in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.
{"title":"Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source","authors":"Rami Masri, Boqian Shen, Beatrice Riviere","doi":"10.1051/m2an/2022095","DOIUrl":"https://doi.org/10.1051/m2an/2022095","url":null,"abstract":"The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a priori error estimates in the L 2 norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the L 2 norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the L 2 norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in L 2 in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136156898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we propose and analyze finite element discontinuous Galerkin methods for the one- and two-dimensional stochastic Maxwell equations with multiplicative noise. The discrete energy law of the semi-discrete DG methods were studied. Optimal error estimate of the semi-discrete method is obtained for the one-dimensional case, and the two-dimensional case on both rectangular meshes and triangular meshes under certain mesh assumptions. Strong Taylor 2.0 scheme is used as the temporal discretization. Both one- and two-dimensional numerical results are presented to validate the theoretical analysis results.
{"title":"Discontinuous Galerkin methods for stochastic Maxwell equations with multiplicative noise","authors":"Jiawei Sun, Chi-Wang Shu, Yulong Xing","doi":"10.1051/m2an/2022084","DOIUrl":"https://doi.org/10.1051/m2an/2022084","url":null,"abstract":"In this paper we propose and analyze finite element discontinuous Galerkin methods for the one- and two-dimensional stochastic Maxwell equations with multiplicative noise. The discrete energy law of the semi-discrete DG methods were studied. Optimal error estimate of the semi-discrete method is obtained for the one-dimensional case, and the two-dimensional case on both rectangular meshes and triangular meshes under certain mesh assumptions. Strong Taylor 2.0 scheme is used as the temporal discretization. Both one- and two-dimensional numerical results are presented to validate the theoretical analysis results.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135907378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a fully practical numerical scheme for the simulation of the stochastic total variation flow (STVF). The approximation is based on a stable time-implicit finite element space-time approximation of a regularized STVF equation. The approximation also involves a finite dimensional discretization of the noise that makes the scheme fully implementable on physical hardware. We show that the proposed numerical scheme converges in law to a solution that is defined in the sense of stochastic variational inequalities (SVIs). Under strengthened assumptions the convergence can be show to holds even in probability. As a by product of our convergence analysis we provide a generalization of the concept of probabilistically weak solutions of stochastic partial differential equation (SPDEs) to the setting of SVIs. We also prove convergence of the numerical scheme to a probabilistically strong solution in probability if pathwise uniqueness holds. We perform numerical simulations to illustrate the behavior of the proposed numerical scheme as well as its non-conforming variant in the context of image denoising.
{"title":"Numerical approximation of probabilistically weak and strong solutions of the stochastic total variation flow","authors":"Lubomir Banas, Martin Ondrejat","doi":"10.1051/m2an/2022089","DOIUrl":"https://doi.org/10.1051/m2an/2022089","url":null,"abstract":"We propose a fully practical numerical scheme for the simulation of the stochastic total variation flow (STVF). The approximation is based on a stable time-implicit finite element space-time approximation of a regularized STVF equation. The approximation also involves a finite dimensional discretization of the noise that makes the scheme fully implementable on physical hardware. We show that the proposed numerical scheme converges in law to a solution that is defined in the sense of stochastic variational inequalities (SVIs). Under strengthened assumptions the convergence can be show to holds even in probability. As a by product of our convergence analysis we provide a generalization of the concept of probabilistically weak solutions of stochastic partial differential equation (SPDEs) to the setting of SVIs. We also prove convergence of the numerical scheme to a probabilistically strong solution in probability if pathwise uniqueness holds. We perform numerical simulations to illustrate the behavior of the proposed numerical scheme as well as its non-conforming variant in the context of image denoising.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"218 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136156902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The numerical approximation of nonsmooth solutions of the semilinear Klein–Gordon equation in the d -dimensional space, with d = 1, 2, 3, is studied based on the discovery of a new cancellation structure in the equation. This cancellation structure allows us to construct a low-regularity correction of the Lie splitting method ( i.e. , exponential Euler method), which can significantly improve the accuracy of the numerical solutions under low-regularity conditions compared with other second-order methods. In particular, the proposed time-stepping method can have second-order convergence in the energy space under the regularity condition $ (u,{mathrm{partial }}_tu)in {L}^{mathrm{infty }}(0,T;{H}^{1+frac{d}{4}}times {H}^{frac{d}{4}})$ . In one dimension, the proposed method is shown to have almost $ frac{4}{3}$ -order convergence in L ∞ (0, T; H 1 × L 2 ) for solutions in the same space, i.e. , no additional regularity in the solution is required. Rigorous error estimates are presented for a fully discrete spectral method with the proposed low-regularity time-stepping scheme. The numerical experiments show that the proposed time-stepping method is much more accurate than previously proposed methods for approximating the time dynamics of nonsmooth solutions of the semilinear Klein–Gordon equation.
{"title":"A second-order low-regularity correction of Lie splitting for the semilinear Klein–Gordon equation","authors":"Buyang Li, Katharina Schratz, Franco Zivcovich","doi":"10.1051/m2an/2022096","DOIUrl":"https://doi.org/10.1051/m2an/2022096","url":null,"abstract":"The numerical approximation of nonsmooth solutions of the semilinear Klein–Gordon equation in the d -dimensional space, with d = 1, 2, 3, is studied based on the discovery of a new cancellation structure in the equation. This cancellation structure allows us to construct a low-regularity correction of the Lie splitting method ( i.e. , exponential Euler method), which can significantly improve the accuracy of the numerical solutions under low-regularity conditions compared with other second-order methods. In particular, the proposed time-stepping method can have second-order convergence in the energy space under the regularity condition $ (u,{mathrm{partial }}_tu)in {L}^{mathrm{infty }}(0,T;{H}^{1+frac{d}{4}}times {H}^{frac{d}{4}})$ . In one dimension, the proposed method is shown to have almost $ frac{4}{3}$ -order convergence in L ∞ (0, T; H 1 × L 2 ) for solutions in the same space, i.e. , no additional regularity in the solution is required. Rigorous error estimates are presented for a fully discrete spectral method with the proposed low-regularity time-stepping scheme. The numerical experiments show that the proposed time-stepping method is much more accurate than previously proposed methods for approximating the time dynamics of nonsmooth solutions of the semilinear Klein–Gordon equation.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"347 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136156906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a quantitative effective medium theory for two types of acoustic metamaterials constituted of a large number N of small heterogeneities of characteristic size s , randomly and independently distributed in a bounded domain. We first consider a “sound-soft” material, in which the total wave field satisfies a Dirichlet boundary condition on the acoustic obstacles. In the “sub-critical” regime sN = O (1), we obtain that the effective medium is governed by a dissipative Lippmann–Schwinger equation which approximates the total field with a relative mean-square error of order O (max(( sN ) 2 N -1/3, N -1/2)). We retrieve the critical size s ~ 1/ N of the literature at which the effects of the obstacles can be modelled by a “strange term” added to the Helmholtz equation. Second, we consider high-contrast acoustic metamaterials, in which each of the N heterogeneities are packets of K inclusions filled with a material of density much lower than the one of the background medium. As the contrast parameter vanishes, δ → 0, the effective medium admits K resonant characteristic sizes ( s i ( δ )) 1≤ i ≤ K and is governed by a Lippmann–Schwinger equation, which is diffusive or dispersive (with negative refractive index) for frequencies ω respectively slightly larger or slightly smaller than the corresponding K resonant frequencies ( ω i ( δ )) 1≤ i ≤ K . These conclusions are obtained under the condition that (i) the resonance is of monopole type, and (ii) lies in the “subcritical regime” where the contrast parameter is small enough, i.e. δ = o ( N −2 )), while the considered frequency is “not too close” to the resonance, i.e. N δ 1/2 = O (|1 - s/s i (δ)|). Our mathematical analysis and the current literature allow us to conjecture that “solidification” phenomena are expected to occur in the “super-critical” regime N δ 1/2 |1 - s/s i (δ)| -1 → + ∞.
本文提出了两类声学超材料的定量有效介质理论，这些材料由大量N个特征尺寸为s的小异质性组成，随机独立分布在有界域中。我们首先考虑一种“声软”材料，其中总波场满足声障碍物上的狄利克雷边界条件。在“次临界”状态sN = O(1)下，我们得到了有效介质由耗散Lippmann-Schwinger方程控制，该方程近似于总场，相对均方误差为O阶(max((sN) 2n -1/3, N -1/2))。我们检索了文献的临界尺寸s ~ 1/ N，在该临界尺寸下，障碍物的影响可以通过在亥姆霍兹方程中添加一个“奇怪项”来建模。其次，我们考虑了高对比度声学超材料，其中每个N非均质都是由密度远低于背景介质的材料填充的K包体包。当对比参数δ→0消失时，有效介质承认K个共振特征尺寸(si (δ)) 1≤i≤K，并受Lippmann-Schwinger方程支配，该方程在频率ω略大于或略小于对应K个共振频率(ω i (δ)) 1≤i≤K时为扩散或色散(具有负折射率)。这些结论是在以下条件下得到的:(i)谐振为单极子型，(ii)处于对比参数足够小的“亚临界区”，即δ = 0 (N−2))，而考虑的频率与谐振“不太接近”，即N δ 1/2 = o (|1 - s/s i (δ)|)。我们的数学分析和目前的文献允许我们推测，“凝固”现象预计将发生在“超临界”状态N δ 1/2 |1 - s/s i (δ)| -1→+∞。
{"title":"Homogenization of sound-soft and high-contrast acoustic metamaterials in subcritical regimes","authors":"Florian Feppon, H. Ammari","doi":"10.1051/m2an/2022098","DOIUrl":"https://doi.org/10.1051/m2an/2022098","url":null,"abstract":"We propose a quantitative effective medium theory for two types of acoustic metamaterials constituted of a large number N of small heterogeneities of characteristic size s , randomly and independently distributed in a bounded domain. We first consider a “sound-soft” material, in which the total wave field satisfies a Dirichlet boundary condition on the acoustic obstacles. In the “sub-critical” regime sN = O (1), we obtain that the effective medium is governed by a dissipative Lippmann–Schwinger equation which approximates the total field with a relative mean-square error of order O (max(( sN ) 2 N -1/3, N -1/2)). We retrieve the critical size s ~ 1/ N of the literature at which the effects of the obstacles can be modelled by a “strange term” added to the Helmholtz equation. Second, we consider high-contrast acoustic metamaterials, in which each of the N heterogeneities are packets of K inclusions filled with a material of density much lower than the one of the background medium. As the contrast parameter vanishes, δ → 0, the effective medium admits K resonant characteristic sizes ( s i ( δ )) 1≤ i ≤ K and is governed by a Lippmann–Schwinger equation, which is diffusive or dispersive (with negative refractive index) for frequencies ω respectively slightly larger or slightly smaller than the corresponding K resonant frequencies ( ω i ( δ )) 1≤ i ≤ K . These conclusions are obtained under the condition that (i) the resonance is of monopole type, and (ii) lies in the “subcritical regime” where the contrast parameter is small enough, i.e. δ = o ( N −2 )), while the considered frequency is “not too close” to the resonance, i.e. N δ 1/2 = O (|1 - s/s i (δ)|). Our mathematical analysis and the current literature allow us to conjecture that “solidification” phenomena are expected to occur in the “super-critical” regime N δ 1/2 |1 - s/s i (δ)| -1 → + ∞.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135742588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Active Flux is a recently developed numerical method for hyperbolic conservation laws. Its classical degrees of freedom are cell averages and point values at cell interfaces. These latter are shared between adjacent cells, leading to a globally continuous reconstruction. The update of the point values includes upwinding, but without solving a Riemann Problem. The update of the cell average requires a flux at the cell interface, which can be immediately obtained using the point values. This paper explores different extensions of Active Flux to arbitrarily high order of accuracy, while maintaining the idea of global continuity. We propose to either increase the stencil while keeping the same degrees of freedom, or to increase the number of point values, or to include higher moments as new degrees of freedom. These extensions have different properties, and reflect different views upon the relation of Active Flux to the families of Finite Volume, Finite Difference and Finite Element methods.
{"title":"Extensions of Active Flux to arbitrary order of accuracy","authors":"Rémi Abgrall, Wasilij Barsukow","doi":"10.1051/m2an/2023004","DOIUrl":"https://doi.org/10.1051/m2an/2023004","url":null,"abstract":"Active Flux is a recently developed numerical method for hyperbolic conservation laws. Its classical degrees of freedom are cell averages and point values at cell interfaces. These latter are shared between adjacent cells, leading to a globally continuous reconstruction. The update of the point values includes upwinding, but without solving a Riemann Problem. The update of the cell average requires a flux at the cell interface, which can be immediately obtained using the point values. This paper explores different extensions of Active Flux to arbitrarily high order of accuracy, while maintaining the idea of global continuity. We propose to either increase the stencil while keeping the same degrees of freedom, or to increase the number of point values, or to include higher moments as new degrees of freedom. These extensions have different properties, and reflect different views upon the relation of Active Flux to the families of Finite Volume, Finite Difference and Finite Element methods.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135907377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Juntao Huang, Thomas Izgin, Stefan Kopecz, Andreas Meister, Chi-Wang Shu
In this paper, we perform a stability analysis for classes of second and third order accurate strong-stability-preserving modified Patankar–Runge–Kutta (SSPMPRK) schemes, which were introduced in Huang and Shu [ J. Sci. Comput. 78 (2019) 1811–1839] and Huang et al . [ J. Sci. Comput. 79 (2019) 1015–1056] and can be used to solve convection equations with stiff source terms, such as reactive Euler equations, with guaranteed positivity under the standard CFL condition due to the convection terms only. The analysis allows us to identify the range of free parameters in these SSPMPRK schemes in order to ensure stability. Numerical experiments are provided to demonstrate the validity of the analysis.
{"title":"On the stability of strong-stability-preserving modified Patankar–Runge–Kutta schemes","authors":"Juntao Huang, Thomas Izgin, Stefan Kopecz, Andreas Meister, Chi-Wang Shu","doi":"10.1051/m2an/2023005","DOIUrl":"https://doi.org/10.1051/m2an/2023005","url":null,"abstract":"In this paper, we perform a stability analysis for classes of second and third order accurate strong-stability-preserving modified Patankar–Runge–Kutta (SSPMPRK) schemes, which were introduced in Huang and Shu [ J. Sci. Comput. 78 (2019) 1811–1839] and Huang et al . [ J. Sci. Comput. 79 (2019) 1015–1056] and can be used to solve convection equations with stiff source terms, such as reactive Euler equations, with guaranteed positivity under the standard CFL condition due to the convection terms only. The analysis allows us to identify the range of free parameters in these SSPMPRK schemes in order to ensure stability. Numerical experiments are provided to demonstrate the validity of the analysis.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"165 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135693485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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