A fast sparse grid based space–time boundary element method for the nonstationary heat equation

AbstractThe paper deals with the numerical solution of 2D wave propagation exterior problems including viscous and material damping coefficients and equipped by Neumann boundary condition, hence modeling the hard scattering of damped waves. The differential problem, which includes, besides diffusion, advection and reaction terms, is written as a space–time boundary integral equation (BIE) whose kernel is given by the hypersingular fundamental solution of the 2D damped waves operator. The resulting BIE is solved by a modified Energetic Boundary Element Method, where a suitable kernel treatment is introduced for the evaluation of the discretization linear system matrix entries represented by space–time quadruple integrals with hypersingular kernel in space variables. A wide variety of numerical results, obtained varying both damping coefficients and discretization parameters, is presented and shows accuracy and stability of the proposed technique, confirming what was theoretically proved for the simpler undamped case. Post-processing phase is also taken into account, giving the approximate solution of the exterior differential problem involving damped waves propagation around disconnected obstacles and bounded domains.

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On numerical methods for constructing benchmark solutions to a nonlinear heat equation with a singularity

Diagnostics Resource and Mechanics of materials and structures ◽

10.17804/2410-9908.2020.5.026-044 ◽

2020 ◽

pp. 26-44

Author(s):

A. L. Kazakov ◽

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L. F. Spevak ◽

E. L. Spevak ◽

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...

Keyword(s):

Boundary Element Method ◽

Finite Difference ◽

Power Series ◽

Heat Equation ◽

Boundary Element ◽

Nonlinear Heat Equation ◽

Cauchy Problems ◽

Wide Range ◽

Benchmark Solutions ◽

Element Method

The paper deals with the construction of exact solutions to a nonlinear heat equation with degeneration in the case of the zero value of the required function. Generically self-similar solutions and traveling wave solutions are considered, the construction of which reduces to solving Cauchy problems for a nonlinear second-order ordinary differential equation with a singularity before the higher derivative. Two approaches are proposed to solve the Cauchy problems: the analytical solution by the power series method and the numerical solution by the boundary element method on a specified segment. A complex computational experiment is carried out to compare the above two methods with each other and with the finite difference methods, namely the Euler method and the fourth-order Runge-Kutta method. Power series segments are used on the first step of the finite difference solutions in order to resolve the singularity. The comparison of the application domains, the accuracy of the solutions and their dependence on the parameters of a certain problem shows that the boundary element method is the most universal, although not the most accurate for some particular examples. The conclusions drawn allow us to construct benchmark solutions to verify the approximate solutions of the nonlinear heat equation by various methods in a wide range of parameter values.

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