Convergence, stability analysis, and solvers for approximating sublinear positone and semipositone boundary value problems using finite difference methods
Convergence, stability analysis, and solvers for approximating sublinear positone and semipositone boundary value problems using finite difference methods
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DOI：
10.1016/j.cam.2021.113880
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发表时间：
2021-10
期刊：
影响因子：
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通讯作者：
T. Lewis;Q. Morris;Yi Zhang
中科院分区：
文献类型：
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作者：
T. Lewis;Q. Morris;Yi Zhang
正、半正边值问题是数学生物学中的反应扩散模型和非线性生热理论中的半线性椭圆型偏微分方程组。在一定条件下，问题可能有多个正解，甚至不存在正解。我们发展了证明次线性问题正解的简单有限差分逼近的可容许性、稳定性和收敛结果的分析方法。我们还开发了保证求解器，它可以检测正音问题的不唯一性和半正音问题的不存在。容许性和稳定性的结果是基于采用通常用于分析潜在偏微分方程组的子解和上解的方法。新的收敛分析技术直接证明了有限差分逼近的所有逐点极限都是边值问题的解，消除了伪代数解困扰方法收敛的可能性。大多数已知的正定和半正定边值问题的逼近方法依赖于打靶法；因此，它们仅限于一维问题和/或径向解。本文的结果将为用简单的逼近方法在更高维和更一般的区域上逼近正、半正边值问题奠定基础。给出了具有多个正解的已知应用问题的数值试验。测试集中于逼近某些正解以及生成离散分支曲线以支持已知的偏微分方程组问题的存在唯一性结果。
Positone and semipositone boundary value problems are semilinear elliptic partial differential equations (PDEs) that arise in reaction–diffusion models in mathematical biology and the theory of nonlinear heat generation. Under certain conditions, the problems may have multiple positive solutions or even nonexistence of a positive solution. We develop analytic techniques for proving admissibility, stability, and convergence results for simple finite difference approximations of positive solutions to sublinear problems. We also develop guaranteed solvers that can detect nonuniqueness for positone problems and nonexistence for semipositone problems. The admissibility and stability results are based on adapting the method of sub- and supersolutions typically used to analyze the underlying PDEs. The new convergence analysis technique directly shows that all pointwise limits of finite difference approximations are solutions to the boundary value problem eliminating the possibility of false algebraic solutions plaguing the convergence of the methods. Most known approximation methods for positone and semipositone boundary value problems rely upon shooting techniques; hence, they are restricted to one-dimensional problems and/or radial solutions. The results in this paper will serve as a foundation for approximating positone and semipositone boundary value problems in higher dimensions and on more general domains using simple approximation methods. Numerical tests for known applied problems with multiple positive solutions are provided. The tests focus on approximating certain positive solutions as well as generating discrete bifurcation curves that support the known existence and uniqueness results for the PDE problem.