Compact Difference Schemes for Linear Partial Differential Equations

<p>Various high-order compact difference schemes constructed for approximation of one-dimensional linear partial differential equations: diffusion equation (including case of variable time-independent diffusion coefficient), modified Schrodinger equation, and rod lateral vibrations equation.</p><p>Classical approach to difference schemes&rsquo; construction is usually (except for splitting methods, variable directions methods and semi-implicit schemes) based on separate algorithms for time and space approximations; compact approach uses unite approximation to all variables. To determine coefficients of compact schemes test functions technique is used: difference relation should be true for simplest pairs of monomials, which satisfies the differential problem. Presented approach to compact schemes&rsquo; construction can be used for numerical solution of various (not only linear) problems of mathematical physics and financial mathematics.</p><p>In particular, here for diffusion equation and Schrodinger equation various types of compact difference schemes were constructed. The most optimal schemes were found by comparison of different properties (including results of numerical experiments). Order of approximation, stability, effectiveness and different types of monotony for various schemes were estimated, and these estimates were validated by numerical experiments. Obtained schemes preserve the first integral for homogeneous Schrodinger and rod lateral vibrations equations with high precision. Compact approximation of different types of boundary conditions for rod lateral vibration equation is also considered.</p><p>Presented schemes can be used as modules of complex computational systems. Obtained results form a basis for the development of compact difference schemes for similar and more difficult problems: complicated equations with high-order derivatives, not necessarily resolved with respect to the highest time derivative, equations with variable coefficients, multidimensional equations and systems of equations.</p>