Helmholtz Equation In Space Research Articles

Balancing truncation and round-off errors in FEM: One-dimensional analysis

In finite element methods, the accuracy of the solution cannot increase indefinitely since the round-off error related to limited computer precision increases when the number of degrees of freedom (DoFs) is large enough. Because a priori information of the highest attainable accuracy is of great interest, we construct an innovative method to obtain the highest attainable accuracy given the order of the elements. In this method, the truncation error is extrapolated when it converges at the asymptotic rate, and the bound of the round-off error follows from a generically valid error estimate, obtained and validated through extensive numerical experiments. The highest attainable accuracy is obtained by minimizing the sum of these two types of errors. We validate this method using a one-dimensional Helmholtz equation in space. It shows that the highest attainable accuracy can be accurately predicted, and the CPU time required is much smaller compared with that using successive grid refinement.

Exact solutions for logistic reaction–diffusion equations in biology

Reaction-diffusion equations with a nonlinear source have been widely used to model various systems, with particular application to biology. Here, we provide a solution technique for these types of equations in $N$-dimensions. The nonclassical symmetry method leads to a single relationship between the nonlinear diffusion coefficient and the nonlinear reaction term; the subsequent solutions for the Kirchhoff variable are exponential in time (either growth or decay) and satisfy the linear Helmholtz equation in space. Example solutions are given in two dimensions for particular parameter sets for both quadratic and cubic reaction terms.

Exact non-classical symmetry solutions of Arrhenius reaction–diffusion

Exact solutions for nonlinear Arrhenius reaction–diffusion are constructed in n -dimensions. A single relationship between nonlinear diffusivity and the nonlinear reaction term leads to a non-classical Lie symmetry whose invariant solutions have a heat flux that is exponential in time (either growth or decay), and satisfying a linear Helmholtz equation in space. This construction also extends to heterogeneous diffusion wherein the nonlinear diffusivity factorizes to the product of a function of temperature and a function of position. Example solutions are given with applications to heat conduction in conjunction with either exothermic or endothermic reactions, and to soil–water flow in conjunction with water extraction by a web of plant roots.

A Legendre spectral element method for the Stefan problem

AbstractIn this paper we present a Legendre spectral element method for solution of multi‐dimensional unsteady change‐of‐phase Stefan problems. The spectral element method is a high‐order (p‐type) finite element technique, in which the computational domain is broken up into general (curved) quadrilateral macroelements, and the solution, data and geometry are expanded within each element in terms of tensor‐product Lagrangian interpolants. The discrete equations are generated by a Galerkin formulation followed by Gauss–Lobatto Legendre quadrature, for which it is shown that exponential convergence to smooth solutions is obtained as the polynomial order of fixed elements is increased. The spectral element equations are inverted by conjugate gradient iteration, in which the matrix‐vector products are calculated efficiently using tensor‐product sum‐factorization.To solve the Stefan problem numerically, the heat equations in the liquid and solid phases are transformed to fixed domains applying an interface‐local time‐dependent immobilization transformation technique. The modified heat equations are discretized using finite differences in time, resulting at each time step in a Helmholtz equation in space that is solved using Legendre spectral element elliptic discretizations. The new interface position is then computed using a variationally consistent flux treatment along the phase boundary, and the solution is projected upon the corresponding updated mesh. The rapid convergence rate and stability of the method are discussed, and numerical results are presented for a one‐dimensional Stefan problem using both a semi‐implicit and a fully implicit time‐stepping scheme. Finally, a two‐dimensional Stefan problem with a complex phase boundary is solved using the semi‐implicit scheme.