An integral equation-based numerical method for the forced heat equation on complex domains

被引:9
作者
Fryklund, Fredrik [1 ]
Kropinski, Mary Catherine A. [2 ]
Tornberg, Anna-Karin [1 ]
机构
[1] KTH Royal Inst Technol, Dept Math, Stockholm, Sweden
[2] Simon Fraser Univ, Dept Math, Burnaby, BC, Canada
基金
瑞典研究理事会;
关键词
Heat equation; Boundary integral method; Modified Helmholtz equation; Yukawa potential; Quadrature; Complex domains; Function extension; Rothe's method; MODIFIED HELMHOLTZ-EQUATION; BOUNDARY-VALUE-PROBLEMS; LAYER POTENTIALS;
D O I
10.1007/s10444-020-09804-z
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Integral equation-based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.
引用
收藏
页数:36
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