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While more complex methods like mesh or multi/adaptive-grid can be used to approximately solve these PDEs, it is instructive to learn about DNS on a lattice - the step size needed to capture the features of the flow accurately can be determined algebraically from the Reynolds number. Inspecting the process of numerically solving PDEs on a lattice, one finds that the problem resolves (in the implicit step case) to solving linear systems - this is typically why linear algebra is introduced in an engineering context. This gets to the essential mathematical features of numerical methods. Some introductory course notes on numerical analysis by Olver are [https://www-users.cse.umn.edu/~olver/num.html here], and a related book of his [https://www.google.com/books/edition/Applied_Linear_Algebra/LfJdDwAAQBAJ?hl=en&gbpv=1 here]. Closer to our direction of structure preserving methods are [https://www-users.cse.umn.edu/~arnold/597.00-01/nabook.pdf Douglas Arnold's Course notes]. His book appears under [[Theory of Elasticity (Book)|Landau 7, Theory of Elasticity]], where numerical methods are continued. Solving linear systems efficiently means, for instance, looking for efficient ways to multiply matrices (e.g. Strassen's algorithm) and to interpolate data. At an even lower level one studied stability of numerical methods or even the implementation of arithmetic on computers in general. Hoping to capture the basics needed and some of the cutting edge here, Landsberg's book explains how one can think of the matrix multiplication function as a tensor itself, and do algebraic geometry in the space of tensors to better understand the complexity of the algorithms involved and to find efficient solutions. A later related book of his on tensor geometry that mentions Hackbusch's motivation is grouped with [[Statistical Physics part 2 - quantum theory (Book)|Landau 9, condensed matter physics]]. Hackbusch's books work from a similar ground of tensor geometry but focuses on the numerical PDE application. His later books develop the cutting edge of the aforementioned multigrid methods. And since interpolation/galerkin/finite element methods involve inherently functional-analytical ideas, quantum mechanics background helps in thinking about the function spaces involved but is not strictly necessary. | While more complex methods like mesh or multi/adaptive-grid can be used to approximately solve these PDEs, it is instructive to learn about DNS on a lattice - the step size needed to capture the features of the flow accurately can be determined algebraically from the Reynolds number. Inspecting the process of numerically solving PDEs on a lattice, one finds that the problem resolves (in the implicit step case) to solving linear systems - this is typically why linear algebra is introduced in an engineering context. This gets to the essential mathematical features of numerical methods. Some introductory course notes on numerical analysis by Olver are [https://www-users.cse.umn.edu/~olver/num.html here], and a related book of his [https://www.google.com/books/edition/Applied_Linear_Algebra/LfJdDwAAQBAJ?hl=en&gbpv=1 here]. Closer to our direction of structure preserving methods are [https://www-users.cse.umn.edu/~arnold/597.00-01/nabook.pdf Douglas Arnold's Course notes]. His book appears under [[Theory of Elasticity (Book)|Landau 7, Theory of Elasticity]], where numerical methods are continued. Solving linear systems efficiently means, for instance, looking for efficient ways to multiply matrices (e.g. Strassen's algorithm) and to interpolate data. At an even lower level one studied stability of numerical methods or even the implementation of arithmetic on computers in general. Hoping to capture the basics needed and some of the cutting edge here, Landsberg's book explains how one can think of the matrix multiplication function as a tensor itself, and do algebraic geometry in the space of tensors to better understand the complexity of the algorithms involved and to find efficient solutions. A later related book of his on tensor geometry that mentions Hackbusch's motivation is grouped with [[Statistical Physics part 2 - quantum theory (Book)|Landau 9, condensed matter physics]]. Hackbusch's books work from a similar ground of tensor geometry but focuses on the numerical PDE application. His later books develop the cutting edge of the aforementioned multigrid methods. And since interpolation/galerkin/finite element methods involve inherently functional-analytical ideas, quantum mechanics background helps in thinking about the function spaces involved but is not strictly necessary. | ||
Numerical methods are still young, Monte Carlo was developed during the Manhattan Project, and software tools stick to rudimentary approaches to rely on high performance computing. Without overwhelming this page with general aspects of numerical algorithms, we will present foundations and other interesting applications [[Numerical Analysis|here]] | Numerical methods are still young, Monte Carlo was developed during the Manhattan Project, and software tools stick to rudimentary approaches to rely on high performance computing. Without overwhelming this page with general aspects of numerical algorithms, we will present foundations and other interesting applications [[Numerical Analysis|here]] As with our previous philosophy, the future is in geometric physics - numerical methods which preserve differential-geometric structures. Starting from the previous numerical and specifically finite-element-mesh ideas, we present some resources for the very active area of differential-form-based interpolation and symplectic integration methods. Their importance cannot be understated, the implementation of conservation laws at the numerical level makes numerical methods not only produce apparently more physically accurate results, but also be more useful theoretically in the description of physics. | ||
=== Applications === | === Applications === | ||