Pub Date : 2025-12-02DOI: 10.1016/j.apnum.2025.11.011
Jikun Zhao , Kangcheng Deng , Chao Wang , Bei Zhang
This paper aims to develop a mixed finite element method for the three-dimensional quad-curl problem with low-order terms. We prove the regularity estimates on the solution to the primal weak problem under the assumption that the domain is a convex polyhedron. Subsequently, we introduce an auxiliary variable to reformulate the original problem as a mixed problem that consists of two curl-curl equations. Based on the regularity estimates, we establish the equivalence between the primal and mixed formulations. In this mixed finite element method, the primal and auxiliary variables are discretized by the Nédélec’s edge elements. We first derive the suboptimal error estimates for the mixed finite element method. In order to prove the optimal convergence, we construct a special projection with some good properties by using the Maxwell equation under the natural boundary condition. Then, by the duality argument, we prove the optimal error estimates for the approximation to the primal solution in the quad-curl equation. The numerical results illustrate the viability and optimal convergence of this method.
{"title":"Error analysis on the mixed finite element method for a quad-curl problem with low-order terms in three dimensions","authors":"Jikun Zhao , Kangcheng Deng , Chao Wang , Bei Zhang","doi":"10.1016/j.apnum.2025.11.011","DOIUrl":"10.1016/j.apnum.2025.11.011","url":null,"abstract":"<div><div>This paper aims to develop a mixed finite element method for the three-dimensional quad-curl problem with low-order terms. We prove the regularity estimates on the solution to the primal weak problem under the assumption that the domain is a convex polyhedron. Subsequently, we introduce an auxiliary variable to reformulate the original problem as a mixed problem that consists of two curl-curl equations. Based on the regularity estimates, we establish the equivalence between the primal and mixed formulations. In this mixed finite element method, the primal and auxiliary variables are discretized by the Nédélec’s edge elements. We first derive the suboptimal error estimates for the mixed finite element method. In order to prove the optimal convergence, we construct a special projection with some good properties by using the Maxwell equation under the natural boundary condition. Then, by the duality argument, we prove the optimal error estimates for the approximation to the primal solution in the quad-curl equation. The numerical results illustrate the viability and optimal convergence of this method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"222 ","pages":"Pages 17-31"},"PeriodicalIF":2.4,"publicationDate":"2025-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145735273","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-25DOI: 10.1016/j.apnum.2025.11.010
Yujian Jiao , Shuaifei Hu , Xiaoxuan Qi
The Brusselator model is a nonlinear reaction-diffusion system that is widely used in the applied sciences. In this study, we investigate a spectral Galerkin proper orthogonal decomposition (SG-POD) method for the two-dimensional Brusselator model subject to homogeneous boundary conditions. We propose a spectral Galerkin (SG) method based on generalized Jacobi polynomials, combined with the Crank-Nicolson scheme for time discretization. We establish the boundedness, generalized stability, and convergence of the proposed method. Furthermore, we develop a SG-POD scheme for the Brusselator model and analyze its stability and convergence. Extensive numerical experiments demonstrate the efficiency of the proposed scheme and show excellent agreement with the theoretical results. The advantages of the proposed approach are as follows: (i) The use of generalized Jacobi polynomials simplifies the theoretical analysis and yields a sparse discrete system. (ii) The numerical solutions obtained by the SG-POD method achieve spectral accuracy in space. (iii) The SG-POD method significantly reduces computational time while maintaining high accuracy.
{"title":"Spectral Galerkin proper orthogonal decomposition method for Brusselator model","authors":"Yujian Jiao , Shuaifei Hu , Xiaoxuan Qi","doi":"10.1016/j.apnum.2025.11.010","DOIUrl":"10.1016/j.apnum.2025.11.010","url":null,"abstract":"<div><div>The Brusselator model is a nonlinear reaction-diffusion system that is widely used in the applied sciences. In this study, we investigate a spectral Galerkin proper orthogonal decomposition (SG-POD) method for the two-dimensional Brusselator model subject to homogeneous boundary conditions. We propose a spectral Galerkin (SG) method based on generalized Jacobi polynomials, combined with the Crank-Nicolson scheme for time discretization. We establish the boundedness, generalized stability, and convergence of the proposed method. Furthermore, we develop a SG-POD scheme for the Brusselator model and analyze its stability and convergence. Extensive numerical experiments demonstrate the efficiency of the proposed scheme and show excellent agreement with the theoretical results. The advantages of the proposed approach are as follows: (i) The use of generalized Jacobi polynomials simplifies the theoretical analysis and yields a sparse discrete system. (ii) The numerical solutions obtained by the SG-POD method achieve spectral accuracy in space. (iii) The SG-POD method significantly reduces computational time while maintaining high accuracy.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"221 ","pages":"Pages 80-107"},"PeriodicalIF":2.4,"publicationDate":"2025-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145616688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-22DOI: 10.1016/j.apnum.2025.11.008
Yongyong Cai , Fenghua Tong
We present a novel structure preserving approximation for solving the Patlak-Keller-Segel equation, combining conventional numerical discretization with a constrained optimization (or projection) based post-processing. To illustrate the idea, we use finite difference with Crank-Nicolson time stepping, followed by a projection step that solves an optimization problem to enforce positivity and mass conservation in the numerical solution. Rigorous error estimates are established with second-order accuracy in both space and time. Numerical experiments support the theoretical results and demonstrate the efficiency of our proposed approach. Extensive numerical tests demonstrate that the positivity preserving and mass conserving properties are crucial in simulating the Patlak-Keller-Segel equation.
{"title":"Positivity preserving and mass conservative projection methods for the Patlak-Keller-Segel equation","authors":"Yongyong Cai , Fenghua Tong","doi":"10.1016/j.apnum.2025.11.008","DOIUrl":"10.1016/j.apnum.2025.11.008","url":null,"abstract":"<div><div>We present a novel structure preserving approximation for solving the Patlak-Keller-Segel equation, combining conventional numerical discretization with a constrained optimization (or projection) based post-processing. To illustrate the idea, we use finite difference with Crank-Nicolson time stepping, followed by a projection step that solves an optimization problem to enforce positivity and mass conservation in the numerical solution. Rigorous error estimates are established with second-order accuracy in both space and time. Numerical experiments support the theoretical results and demonstrate the efficiency of our proposed approach. Extensive numerical tests demonstrate that the positivity preserving and mass conserving properties are crucial in simulating the Patlak-Keller-Segel equation.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"221 ","pages":""},"PeriodicalIF":2.4,"publicationDate":"2025-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145616690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-20DOI: 10.1016/j.apnum.2025.11.009
Qiwei Feng , Bin Han
<div><div>In this paper, we investigate 1D elliptic equations <span><math><mrow><mo>−</mo><mi>∇</mi><mo>·</mo><mo>(</mo><mi>a</mi><mi>∇</mi><mi>u</mi><mo>)</mo><mo>=</mo><mi>f</mi></mrow></math></span> with rough diffusion coefficients <em>a</em> satisfying 0 < <em>a</em><sub>min</sub> ≤ <em>a</em> ≤ <em>a</em><sub>max</sub> < ∞ and rough source terms <em>f</em> ∈ <em>L</em><sub>2</sub>(Ω). To achieve an accurate and robust numerical solution on a coarse mesh of size <em>H</em>, we introduce a derivative-orthogonal wavelet-based framework. This approach incorporates both regular and specialized basis functions constructed through a novel technique, defining a basis function space that enables effective approximation. We develop a derivative-orthogonal wavelet multiscale method tailored for this framework, proving that the condition number <em>κ</em> of the stiffness matrix satisfies <em>κ</em> ≤ <em>a</em><sub>max</sub>/<em>a</em><sub>min</sub>, independent of <em>H</em>. For the error analysis, we establish that the energy and <em>L</em><sub>2</sub>-norm errors of our method converge at first-order and second-order rates, respectively, for any coarse mesh <em>H</em>. Specifically, the energy and <em>L</em><sub>2</sub>-norm errors are bounded by <span><math><mrow><mn>2</mn><msubsup><mi>a</mi><mrow><mi>min</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup><msub><mrow><mo>∥</mo><mi>f</mi><mo>∥</mo></mrow><mrow><msub><mi>L</mi><mn>2</mn></msub><mrow><mo>(</mo><mstyle><mi>Ω</mi></mstyle><mo>)</mo></mrow></mrow></msub><mi>H</mi></mrow></math></span> and <span><math><mrow><mn>4</mn><msubsup><mi>a</mi><mrow><mi>min</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mrow><mo>∥</mo><mi>f</mi><mo>∥</mo></mrow><mrow><msub><mi>L</mi><mn>2</mn></msub><mrow><mo>(</mo><mstyle><mi>Ω</mi></mstyle><mo>)</mo></mrow></mrow></msub><msup><mi>H</mi><mn>2</mn></msup></mrow></math></span>. Moreover, the numerical approximated solution also possesses the interpolation property at all grid points. We present a range of challenging test cases with continuous, discontinuous, high-frequency, and high-contrast coefficients <em>a</em> to evaluate errors in <em>u, u</em>′ and <em>au</em>′ in both <em>l</em><sub>2</sub> and <em>l</em><sub>∞</sub> norms. We also provide a numerical example that both coefficient <em>a</em> and source term <em>f</em> contain discontinuous, high-frequency and high-contrast oscillations. Additionally, we compare our method with the standard second-order finite element method (FEM) and the special FEM in [6] to assess error behaviors and condition numbers when the mesh is not fine enough to resolve coefficient oscillations. Numerical results confirm the bounded condition numbers and convergence rates, affirming the effectiveness of our approach. Thus, our method is capable of handling both the rough diffusion coefficient <em>a</em> and the rough source term <em>f</em>. In the special case that <em>a</em> is const
{"title":"A derivative-orthogonal wavelet multiscale method for elliptic equations with rough diffusion coefficients","authors":"Qiwei Feng , Bin Han","doi":"10.1016/j.apnum.2025.11.009","DOIUrl":"10.1016/j.apnum.2025.11.009","url":null,"abstract":"<div><div>In this paper, we investigate 1D elliptic equations <span><math><mrow><mo>−</mo><mi>∇</mi><mo>·</mo><mo>(</mo><mi>a</mi><mi>∇</mi><mi>u</mi><mo>)</mo><mo>=</mo><mi>f</mi></mrow></math></span> with rough diffusion coefficients <em>a</em> satisfying 0 < <em>a</em><sub>min</sub> ≤ <em>a</em> ≤ <em>a</em><sub>max</sub> < ∞ and rough source terms <em>f</em> ∈ <em>L</em><sub>2</sub>(Ω). To achieve an accurate and robust numerical solution on a coarse mesh of size <em>H</em>, we introduce a derivative-orthogonal wavelet-based framework. This approach incorporates both regular and specialized basis functions constructed through a novel technique, defining a basis function space that enables effective approximation. We develop a derivative-orthogonal wavelet multiscale method tailored for this framework, proving that the condition number <em>κ</em> of the stiffness matrix satisfies <em>κ</em> ≤ <em>a</em><sub>max</sub>/<em>a</em><sub>min</sub>, independent of <em>H</em>. For the error analysis, we establish that the energy and <em>L</em><sub>2</sub>-norm errors of our method converge at first-order and second-order rates, respectively, for any coarse mesh <em>H</em>. Specifically, the energy and <em>L</em><sub>2</sub>-norm errors are bounded by <span><math><mrow><mn>2</mn><msubsup><mi>a</mi><mrow><mi>min</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup><msub><mrow><mo>∥</mo><mi>f</mi><mo>∥</mo></mrow><mrow><msub><mi>L</mi><mn>2</mn></msub><mrow><mo>(</mo><mstyle><mi>Ω</mi></mstyle><mo>)</mo></mrow></mrow></msub><mi>H</mi></mrow></math></span> and <span><math><mrow><mn>4</mn><msubsup><mi>a</mi><mrow><mi>min</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mrow><mo>∥</mo><mi>f</mi><mo>∥</mo></mrow><mrow><msub><mi>L</mi><mn>2</mn></msub><mrow><mo>(</mo><mstyle><mi>Ω</mi></mstyle><mo>)</mo></mrow></mrow></msub><msup><mi>H</mi><mn>2</mn></msup></mrow></math></span>. Moreover, the numerical approximated solution also possesses the interpolation property at all grid points. We present a range of challenging test cases with continuous, discontinuous, high-frequency, and high-contrast coefficients <em>a</em> to evaluate errors in <em>u, u</em>′ and <em>au</em>′ in both <em>l</em><sub>2</sub> and <em>l</em><sub>∞</sub> norms. We also provide a numerical example that both coefficient <em>a</em> and source term <em>f</em> contain discontinuous, high-frequency and high-contrast oscillations. Additionally, we compare our method with the standard second-order finite element method (FEM) and the special FEM in [6] to assess error behaviors and condition numbers when the mesh is not fine enough to resolve coefficient oscillations. Numerical results confirm the bounded condition numbers and convergence rates, affirming the effectiveness of our approach. Thus, our method is capable of handling both the rough diffusion coefficient <em>a</em> and the rough source term <em>f</em>. In the special case that <em>a</em> is const","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"221 ","pages":"Pages 108-134"},"PeriodicalIF":2.4,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145682017","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-17DOI: 10.1016/j.apnum.2025.11.005
Helena Biščević , Raffaele D’Ambrosio
The paper is focused on the numerical solution of stochastic reaction-diffusion problems. A special attention is addressed to the conservation of mean-square dissipativity in the time integration of the spatially discretized problem, obtained through finite differences. The analysis highlights the conservative ability of stochastic θ-methods and stochastic θ-IMEX methods, emphasizing the roles of spatial and temporal stepsizes. A selection of numerical experiments is provided, confirming the theoretical expectations.
{"title":"Time integration of dissipative stochastic PDEs","authors":"Helena Biščević , Raffaele D’Ambrosio","doi":"10.1016/j.apnum.2025.11.005","DOIUrl":"10.1016/j.apnum.2025.11.005","url":null,"abstract":"<div><div>The paper is focused on the numerical solution of stochastic reaction-diffusion problems. A special attention is addressed to the conservation of mean-square dissipativity in the time integration of the spatially discretized problem, obtained through finite differences. The analysis highlights the conservative ability of stochastic <em>θ</em>-methods and stochastic <em>θ</em>-IMEX methods, emphasizing the roles of spatial and temporal stepsizes. A selection of numerical experiments is provided, confirming the theoretical expectations.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"222 ","pages":"Pages 1-16"},"PeriodicalIF":2.4,"publicationDate":"2025-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145665666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-14DOI: 10.1016/j.apnum.2025.11.007
Ziheng Chen , Jiao Liu , Anxin Wu
This paper investigates the strong and weak convergence orders of numerical methods for stochastic differential equations (SDEs) driven by time-changed Lévy noise, which effectively model systems subject to non-uniform temporal random perturbations, such as time-varying volatility in financial markets. We first consider the stochastic θ method with θ ∈ [0, 1] for approximating the corresponding non-time-changed SDEs. By employing the duality theorem that links time-changed and non-time-changed SDEs, together with a discrete approximation of the time-change process, we prove that the considered method achieves a strong convergence rate of order 1/2 under global Lipschitz conditions. Furthermore, the Euler–Maruyama method (the case ) is analyzed for weak convergence. Based on the Kolmogorov backward partial integro-differential equation and high-order moment estimates, we establish a weak convergence rate of order 1 for smooth test functions with polynomial growth. Theoretical findings are supported by a series of numerical experiments involving α-stable subordinators and their inverse processes. Both convergence rates are shown to be optimal, consistent with those for Lévy-driven and Brownian-motion-driven SDEs. The proposed framework provides reliable and efficient numerical tools for time-changed Lévy-driven SDEs in applied contexts.
{"title":"Strong and weak convergence orders of numerical methods for SDEs driven by time-changed Lévy noise","authors":"Ziheng Chen , Jiao Liu , Anxin Wu","doi":"10.1016/j.apnum.2025.11.007","DOIUrl":"10.1016/j.apnum.2025.11.007","url":null,"abstract":"<div><div>This paper investigates the strong and weak convergence orders of numerical methods for stochastic differential equations (SDEs) driven by time-changed Lévy noise, which effectively model systems subject to non-uniform temporal random perturbations, such as time-varying volatility in financial markets. We first consider the stochastic <em>θ</em> method with <em>θ</em> ∈ [0, 1] for approximating the corresponding non-time-changed SDEs. By employing the duality theorem that links time-changed and non-time-changed SDEs, together with a discrete approximation of the time-change process, we prove that the considered method achieves a strong convergence rate of order 1/2 under global Lipschitz conditions. Furthermore, the Euler–Maruyama method (the case <span><math><mrow><mi>θ</mi><mo>=</mo><mn>0</mn></mrow></math></span>) is analyzed for weak convergence. Based on the Kolmogorov backward partial integro-differential equation and high-order moment estimates, we establish a weak convergence rate of order 1 for smooth test functions with polynomial growth. Theoretical findings are supported by a series of numerical experiments involving <em>α</em>-stable subordinators and their inverse processes. Both convergence rates are shown to be optimal, consistent with those for Lévy-driven and Brownian-motion-driven SDEs. The proposed framework provides reliable and efficient numerical tools for time-changed Lévy-driven SDEs in applied contexts.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"221 ","pages":"Pages 46-62"},"PeriodicalIF":2.4,"publicationDate":"2025-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145616689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-13DOI: 10.1016/j.apnum.2025.11.006
Yong Chen
This paper proposes and analyzes a uniformly convergent fourth-order compact finite difference scheme for the Robin boundary parabolic partial differential equation (PDE) system arising from lookback option pricing with regime-switching. First, we discretize the problem for the interior computational region by the Crank-Nicolson compact finite difference scheme with truncation errors , where Δτ and Δz are time step size and spatial mesh size respectively. To achieve the global fourth-order convergence over the whole spatial computational region, we establish the Crank-Nicolson compact scheme with truncation errors for the Robin boundary conditions. Under a mild condition that the spatial mesh size Δz is small enough, the global convergence rates are rigorously proved in L∞ norm by the energy method. Finally, several numerical examples are provided to illustrate the theoretical results and show the efficacy of the proposed scheme.
{"title":"Uniformly convergent compact difference scheme for robin boundary parabolic system arising in lookback option pricing with regime-switching","authors":"Yong Chen","doi":"10.1016/j.apnum.2025.11.006","DOIUrl":"10.1016/j.apnum.2025.11.006","url":null,"abstract":"<div><div>This paper proposes and analyzes a uniformly convergent fourth-order compact finite difference scheme for the Robin boundary parabolic partial differential equation (PDE) system arising from lookback option pricing with regime-switching. First, we discretize the problem for the interior computational region by the Crank-Nicolson compact finite difference scheme with truncation errors <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mstyle><mi>Δ</mi></mstyle><msup><mi>τ</mi><mn>2</mn></msup><mo>+</mo><mstyle><mi>Δ</mi></mstyle><msup><mi>z</mi><mn>4</mn></msup><mo>)</mo></mrow></mrow></math></span>, where Δ<em>τ</em> and Δ<em>z</em> are time step size and spatial mesh size respectively. To achieve the global fourth-order convergence over the whole spatial computational region, we establish the Crank-Nicolson compact scheme with truncation errors <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mstyle><mi>Δ</mi></mstyle><msup><mi>τ</mi><mn>2</mn></msup><mo>+</mo><mstyle><mi>Δ</mi></mstyle><msup><mi>z</mi><mn>3</mn></msup><mo>)</mo></mrow></mrow></math></span> for the Robin boundary conditions. Under a mild condition that the spatial mesh size Δ<em>z</em> is small enough, the global convergence rates <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mstyle><mi>Δ</mi></mstyle><msup><mi>τ</mi><mn>2</mn></msup><mo>+</mo><mstyle><mi>Δ</mi></mstyle><msup><mi>z</mi><mn>4</mn></msup><mo>)</mo></mrow></mrow></math></span> are rigorously proved in <em>L</em><sup>∞</sup> norm by the energy method. Finally, several numerical examples are provided to illustrate the theoretical results and show the efficacy of the proposed scheme.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"221 ","pages":"Pages 1-17"},"PeriodicalIF":2.4,"publicationDate":"2025-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145570809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-11DOI: 10.1016/j.apnum.2025.11.002
Adérito Araújo, Diogo Cotrim
The Swift-Hohenberg equation (SH-PDE) is a fundamental model for pattern formation in nonlinear systems with symmetry breaking instabilities. This work presents a semi-implicit finite difference scheme for solving the SH-PDE, taking advantage of a linear/nonlinear decomposition to optimise stability and computational efficiency. We rigorously establish the theoretical properties of the method, including bounds and error estimates, proving stability and convergence under appropriate conditions. Numerical experiments confirm these conclusions, demonstrating second-order spatial accuracy and first-order temporal accuracy. The method is tested under various initial conditions and nonlinearities, capturing characteristic patterns such as stripes, rolls and dots, in line with the expected behaviour of SH-PDE. These results emphasise the robustness and efficiency of the proposed approach, positioning it as a powerful tool for studying pattern formation in nonlinear systems.
{"title":"A semi-implicit finite difference approach for the swift hohenberg equation: Stability, convergence, and pattern formation","authors":"Adérito Araújo, Diogo Cotrim","doi":"10.1016/j.apnum.2025.11.002","DOIUrl":"10.1016/j.apnum.2025.11.002","url":null,"abstract":"<div><div>The Swift-Hohenberg equation (SH-PDE) is a fundamental model for pattern formation in nonlinear systems with symmetry breaking instabilities. This work presents a semi-implicit finite difference scheme for solving the SH-PDE, taking advantage of a linear/nonlinear decomposition to optimise stability and computational efficiency. We rigorously establish the theoretical properties of the method, including bounds and error estimates, proving stability and convergence under appropriate conditions. Numerical experiments confirm these conclusions, demonstrating second-order spatial accuracy and first-order temporal accuracy. The method is tested under various initial conditions and nonlinearities, capturing characteristic patterns such as stripes, rolls and dots, in line with the expected behaviour of SH-PDE. These results emphasise the robustness and efficiency of the proposed approach, positioning it as a powerful tool for studying pattern formation in nonlinear systems.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 373-383"},"PeriodicalIF":2.4,"publicationDate":"2025-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145576256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-11DOI: 10.1016/j.apnum.2025.11.003
Utku Erdoğan , Gabriel Lord
In this paper, we develop numerical methods for solving stochastic differential equations with solutions that evolve within a hypercube D in . Our approach is based on a convex combination of two numerical flows, both of which are constructed from positivity preserving methods. The strong convergence of the Euler version of the method is proven to be of order , and numerical examples are provided to demonstrate that, in some cases, first-order convergence is observed in practice. We compare the Euler and Milstein versions of these new methods to existing domain preservation methods in the literature and observe our methods are robust, more widely applicable and that, in most cases, have a smaller error constant.
{"title":"Preserving invariant domains and strong approximation of stochastic differential equations","authors":"Utku Erdoğan , Gabriel Lord","doi":"10.1016/j.apnum.2025.11.003","DOIUrl":"10.1016/j.apnum.2025.11.003","url":null,"abstract":"<div><div>In this paper, we develop numerical methods for solving stochastic differential equations with solutions that evolve within a hypercube <em>D</em> in <span><math><msup><mi>R</mi><mi>d</mi></msup></math></span>. Our approach is based on a convex combination of two numerical flows, both of which are constructed from positivity preserving methods. The strong convergence of the Euler version of the method is proven to be of order <span><math><mstyle><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></math></span>, and numerical examples are provided to demonstrate that, in some cases, first-order convergence is observed in practice. We compare the Euler and Milstein versions of these new methods to existing domain preservation methods in the literature and observe our methods are robust, more widely applicable and that, in most cases, have a smaller error constant.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"221 ","pages":"Pages 30-45"},"PeriodicalIF":2.4,"publicationDate":"2025-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145616684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-08DOI: 10.1016/j.apnum.2025.11.001
Changlun Ye , Hai Bi , Liangkun Xu , Xianbing Luo
In this paper, for the Cahn-Hilliard equation with dynamic boundary conditions, we establish a variational time stepping numerical scheme integrated with finite element methods. This scheme is a structure-preserving scheme, which effectively maintains the inherent physical properties of the continuous model including mass conservation and energy dissipation. We demonstrate the existence of discrete solutions without restrictions on the discretization parameters, and establish the uniqueness under mild conditions. Finally, we present ample numerical results which validate our theoretical findings and demonstrate that our numerical scheme can achieve second-order convergence in time. We also apply our scheme to the KLS (proposed by P. Knopf, K.F. Lam, and J. Stange) and KLLM (proposed by P. Knopf, K. F. Lam, C. Liu, and S. Metzger) models, two other Cahn-Hilliard models with dynamic boundaries, and verify that the solutions of KLS model converge to the solutions of KLLM model numerically.
本文针对具有动态边界条件的Cahn-Hilliard方程，建立了与有限元法相结合的变分时步数值格式。该方案是一种结构保持方案，有效地保持了连续模型固有的物理性质，包括质量守恒和能量耗散。我们证明了不受离散化参数限制的离散解的存在性，并在温和条件下证明了其唯一性。最后，我们给出了大量的数值结果来验证我们的理论发现，并证明了我们的数值格式在时间上可以达到二阶收敛。我们还将我们的方案应用于KLS （P. Knopf， K.F. Lam, and J. Stange提出）和KLLM （P. Knopf， K.F. Lam， C. Liu, and S. Metzger提出）模型以及另外两种具有动态边界的Cahn-Hilliard模型，并在数值上验证了KLS模型的解收敛于KLLM模型的解。
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