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This subtopic introduces the core ideas in numerical methods for pdes, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.
Finite difference methods for initial value and initial-boundary value problems involving PDEs. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Wikipedia: Finite difference method
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.
Finite volume methods for initial value and initial-boundary value problems involving PDEs. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Wikipedia: Finite volume method
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.
Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Wikipedia: Numerical stability
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.
Error bounds for numerical methods for initial value and initial-boundary value problems. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.
Method of lines for initial value and initial-boundary value problems involving PDEs. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.
Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.
Method of characteristics for initial value and initial-boundary value problems involving PDEs. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Wikipedia: Method of characteristics
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.
Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.
Inverse problems for initial value and initial-boundary value problems involving PDEs. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.
Boundary element methods for initial value and initial-boundary value problems involving PDEs. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Wikipedia: Boundary element method
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.
Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Wikipedia: Adaptive mesh refinement
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.
Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.
Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.
Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.
Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.
Fundamental solutions, Green's function methods, etc. for initial value and initial-boundary value problems involving PDEs. This topic deals with partial differential equations and focuses on the numerical and analytical challenges of solving problems in this setting with reliable, efficient computational methods.
Used to build reliable computational methods for partial differential equations that arise in science, engineering, and data-driven modeling.