High order accurate methods for the evaluation of layer heat potentials. (English) Zbl 1204.65117
Summary: We discuss the numerical evaluation of single and double layer heat potentials in two dimensions on stationary and moving boundaries. One of the principal difficulties in designing high order methods concerns the local behavior of the heat kernel, which is both weakly singular in time and rapidly decaying in space. We show that standard quadrature schemes suffer from a poorly recognized form of inaccuracy, which we refer to as “geometrically induced stiffness,” but that rules based on product integration of the full heat kernel in time are robust. When combined with previously developed fast algorithms for the evolution of the “history part” of layer potentials, diffusion processes in complex, moving geometries can be computed accurately and in nearly optimal time.
MSC:
65M38 | Boundary element methods for initial value and initial-boundary value problems involving PDEs |
35K05 | Heat equation |
65R20 | Numerical methods for integral equations |
31A10 | Integral representations, integral operators, integral equations methods in two dimensions |
65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
45D05 | Volterra integral equations |
45L05 | Theoretical approximation of solutions to integral equations |