Pub Date : 2026-04-01Epub Date: 2025-12-25DOI: 10.1016/j.apnum.2025.12.007
Jiachen Jin, Kangkang Deng, Boyu Wang, Hongxia Wang
Stochastic alternating direction method of multipliers (SADMM) is a popular method for solving nonconvex nonsmooth optimization in various applications. However, it typically requires an empirical selection of the static batch size for gradient estimation, resulting in a challenging trade-off between variance reduction and computational cost. This paper proposes adaptive batch size SADMM, a practical method that dynamically adjusts the batch size based on accumulated differences along the optimization path. We develop a simple convergence analysis to handle the dependence of batch size adaptation that matches the best-known complexity with flexible parameter choices. We further extend this adaptive scheme to reduce the overall complexity of the popular variance-reduced methods, SVRG-ADMM and SPIDER-ADMM. Numerical results validate the effectiveness of our proposed methods.
{"title":"Stochastic ADMM with batch size adaptation for nonconvex nonsmooth optimization","authors":"Jiachen Jin, Kangkang Deng, Boyu Wang, Hongxia Wang","doi":"10.1016/j.apnum.2025.12.007","DOIUrl":"10.1016/j.apnum.2025.12.007","url":null,"abstract":"<div><div>Stochastic alternating direction method of multipliers (SADMM) is a popular method for solving nonconvex nonsmooth optimization in various applications. However, it typically requires an empirical selection of the static batch size for gradient estimation, resulting in a challenging trade-off between variance reduction and computational cost. This paper proposes adaptive batch size SADMM, a practical method that dynamically adjusts the batch size based on accumulated differences along the optimization path. We develop a simple convergence analysis to handle the dependence of batch size adaptation that matches the best-known complexity with flexible parameter choices. We further extend this adaptive scheme to reduce the overall complexity of the popular variance-reduced methods, SVRG-ADMM and SPIDER-ADMM. Numerical results validate the effectiveness of our proposed methods.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"222 ","pages":"Pages 87-107"},"PeriodicalIF":2.4,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145881038","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 : 2026-04-01Epub 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":"2026-04-01","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 : 2026-04-01Epub Date: 2025-12-24DOI: 10.1016/j.apnum.2025.12.005
Chun Wang
In this paper, a new technique of Adomian decomposition method (ADM) for the numerical solutions of nonlinear differential equations and mathematical models is presented. Different from the traditional ADM and LADM (Laplace Adomian decomposition method), the main idea of this new technique is to properly divide the interval where the problem is to be into several subintervals and then apply ADM on each subinterval, respectively. When applying ADM on the latter subinterval, we take the approximate solution on the former subinterval as the initial value so that we can get the numerical solutions at the nodes of these subintervals by calculating in turn. The new technique of ADM has higher precision than the original and traditional ADM and LADM. By using this technique, the accuracy of the numerical solutions is greatly improved. An obvious advantage of this technique is that the precision of the numerical solutions can be finely adjusted according to the actual needs. On the other hand, compared with the traditional method, the errors of the numerical solutions obtained by this technique have a very strong stability.
{"title":"A new technique of ADM to improve the precision and stability of numerical solutions","authors":"Chun Wang","doi":"10.1016/j.apnum.2025.12.005","DOIUrl":"10.1016/j.apnum.2025.12.005","url":null,"abstract":"<div><div>In this paper, a new technique of Adomian decomposition method (ADM) for the numerical solutions of nonlinear differential equations and mathematical models is presented. Different from the traditional ADM and LADM (Laplace Adomian decomposition method), the main idea of this new technique is to properly divide the interval where the problem is to be into several subintervals and then apply ADM on each subinterval, respectively. When applying ADM on the latter subinterval, we take the approximate solution on the former subinterval as the initial value so that we can get the numerical solutions at the nodes of these subintervals by calculating in turn. The new technique of ADM has higher precision than the original and traditional ADM and LADM. By using this technique, the accuracy of the numerical solutions is greatly improved. An obvious advantage of this technique is that the precision of the numerical solutions can be finely adjusted according to the actual needs. On the other hand, compared with the traditional method, the errors of the numerical solutions obtained by this technique have a very strong stability.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"222 ","pages":"Pages 53-67"},"PeriodicalIF":2.4,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145881039","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 : 2026-04-01Epub Date: 2025-12-25DOI: 10.1016/j.apnum.2025.12.003
Hao Zhang , Kexin Li , Wenlin Qiu
In this work, we study two-dimensional diffusion-wave equations with variable exponent, modeling mechanical diffusive wave propagation in viscoelastic media with spatially varying properties. We first transform the diffusion-wave model into an equivalent form via the convolution method. Two time discretization strategies are then applied to approximate each term in the transformed equation, yielding two fully discrete schemes based on a spatial compact finite difference method. To reduce computational cost, the alternating direction implicit (ADI) technique is employed. We prove that both ADI compact schemes are unconditionally stable and convergent. The error estimates established under reasonable regularity assumption, state that the first scheme achieves α(0)-order accuracy in time and fourth-order accuracy in space, while the second scheme attains second-order accuracy in time and fourth-order accuracy in space. Numerical experiments confirm the theoretical predictions and demonstrate the efficiency of the proposed methods.
{"title":"Two ADI compact difference methods for variable-exponent diffusion wave equations","authors":"Hao Zhang , Kexin Li , Wenlin Qiu","doi":"10.1016/j.apnum.2025.12.003","DOIUrl":"10.1016/j.apnum.2025.12.003","url":null,"abstract":"<div><div>In this work, we study two-dimensional diffusion-wave equations with variable exponent, modeling mechanical diffusive wave propagation in viscoelastic media with spatially varying properties. We first transform the diffusion-wave model into an equivalent form via the convolution method. Two time discretization strategies are then applied to approximate each term in the transformed equation, yielding two fully discrete schemes based on a spatial compact finite difference method. To reduce computational cost, the alternating direction implicit (ADI) technique is employed. We prove that both ADI compact schemes are unconditionally stable and convergent. The error estimates established under reasonable regularity assumption, state that the first scheme achieves <em>α</em>(0)-order accuracy in time and fourth-order accuracy in space, while the second scheme attains second-order accuracy in time and fourth-order accuracy in space. Numerical experiments confirm the theoretical predictions and demonstrate the efficiency of the proposed methods.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"222 ","pages":"Pages 68-86"},"PeriodicalIF":2.4,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145881040","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 : 2026-03-01Epub Date: 2025-11-06DOI: 10.1016/j.apnum.2025.10.017
K. Mustapha, W. Mclean, J. Dick, Q.T. Le Gia
Due to the divergence-instability, the accuracy of low-order conforming finite element methods (FEMs) for nearly incompressible elasticity equations deteriorates as the Lamé parameter λ → ∞, or equivalently as the Poisson ratio ν → 1/2. This effect is known as locking or non-robustness. For the piecewise linear case, the error in the L2(Ω)-norm of the standard Galerkin conforming FEM is bounded by Cλh2, resulting in poor accuracy for practical values of h if λ is sufficiently large. In this paper, we show that the locking phenomenon can be reduced by replacing λ with or in the stiffness matrix, where μ is the second Lamé parameter and L is the diameter of the body Ω. We prove that with this modification, the error in the L2(Ω)-norm is bounded by Ch for a constant C that does not depend on λ. Numerical experiments confirm this convergence behaviour and show that, for practical meshes, our method is more accurate than the standard method if λ is larger than about μL/h. Our analysis also shows that the error in the H1(Ω)-norm is bounded by , which improves the Cλ1/2h estimate for the case of conforming FEM.
{"title":"A simple modification to mitigate locking in conforming FEM for nearly incompressible elasticity","authors":"K. Mustapha, W. Mclean, J. Dick, Q.T. Le Gia","doi":"10.1016/j.apnum.2025.10.017","DOIUrl":"10.1016/j.apnum.2025.10.017","url":null,"abstract":"<div><div>Due to the divergence-instability, the accuracy of low-order conforming finite element methods (FEMs) for nearly incompressible elasticity equations deteriorates as the Lamé parameter <em>λ</em> → ∞, or equivalently as the Poisson ratio <em>ν</em> → 1/2. This effect is known as <em>locking</em> or <em>non-robustness</em>. For the piecewise linear case, the error in the <strong>L</strong><sup>2</sup>(Ω)-norm of the standard Galerkin conforming FEM is bounded by <em>Cλh</em><sup>2</sup>, resulting in poor accuracy for practical values of <em>h</em> if <em>λ</em> is sufficiently large. In this paper, we show that the locking phenomenon can be reduced by replacing <em>λ</em> with <span><math><mrow><msub><mi>λ</mi><mi>h</mi></msub><mo>=</mo><mi>λ</mi><mi>μ</mi><mo>/</mo><msqrt><mrow><msup><mi>μ</mi><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mi>λ</mi><mi>h</mi><mo>/</mo><mi>L</mi><mo>)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo><</mo><mi>λ</mi></mrow></math></span> or <span><math><mrow><msub><mi>λ</mi><mi>h</mi></msub><mo>=</mo><mi>λ</mi><mi>μ</mi><mo>/</mo><mrow><mo>(</mo><mi>μ</mi><mo>+</mo><mi>λ</mi><mi>h</mi><mo>/</mo><mi>L</mi><mo>)</mo></mrow><mo><</mo><mi>λ</mi></mrow></math></span> in the stiffness matrix, where <em>μ</em> is the second Lamé parameter and <em>L</em> is the diameter of the body Ω. We prove that with this modification, the error in the <strong>L</strong><sup>2</sup>(Ω)-norm is bounded by <em>Ch</em> for a constant <em>C</em> that does not depend on <em>λ</em>. Numerical experiments confirm this convergence behaviour and show that, for practical meshes, our method is more accurate than the standard method if <em>λ</em> is larger than about <em>μL</em>/<em>h</em>. Our analysis also shows that the error in the <strong>H</strong><sup>1</sup>(Ω)-norm is bounded by <span><math><mrow><mi>C</mi><msubsup><mi>λ</mi><mi>h</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msubsup><mspace></mspace><mi>h</mi></mrow></math></span>, which improves the <em>Cλ</em><sup>1/2</sup> <em>h</em> estimate for the case of conforming FEM.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"221 ","pages":"Pages 18-29"},"PeriodicalIF":2.4,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571207","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 : 2026-03-01Epub 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":"2026-03-01","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 : 2026-03-01Epub 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":"2026-03-01","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 : 2026-03-01Epub 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":"2026-03-01","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 : 2026-03-01Epub 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":"2026-03-01","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 : 2026-03-01Epub 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":"2026-03-01","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}
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