Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are ...Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.展开更多
We propose a new high order accurate nodal discontinuous Galerkin(DG)method for the solution of nonlinear hyperbolic systems of partial differential equations(PDE)on unstructured polygonal Voronoi meshes.Rather than u...We propose a new high order accurate nodal discontinuous Galerkin(DG)method for the solution of nonlinear hyperbolic systems of partial differential equations(PDE)on unstructured polygonal Voronoi meshes.Rather than using classical polynomials of degree N inside each element,in our new approach the discrete solution is represented by piecewise continuous polynomials of degree N within each Voronoi element,using a continuous finite element basis defined on a subgrid inside each polygon.We call the resulting subgrid basis an agglomerated finite element(AFE)basis for the DG method on general polygons,since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles.The basis functions on each sub-triangle are defined,as usual,on a universal reference element,hence allowing to compute universal mass,flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only.Consequently,the construction of an efficient quadrature-free algorithm is possible,despite the unstructured nature of the computational grid.High order of accuracy in time is achieved thanks to the ADER approach,making use of an element-local space-time Galerkin finite element predictor.The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations.The numerical results have been checked with reference solutions available in literature and also systematically compared,in terms of computational efficiency and accuracy,with those obtained by the corresponding modal DG version of the scheme.展开更多
文摘Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.
文摘We propose a new high order accurate nodal discontinuous Galerkin(DG)method for the solution of nonlinear hyperbolic systems of partial differential equations(PDE)on unstructured polygonal Voronoi meshes.Rather than using classical polynomials of degree N inside each element,in our new approach the discrete solution is represented by piecewise continuous polynomials of degree N within each Voronoi element,using a continuous finite element basis defined on a subgrid inside each polygon.We call the resulting subgrid basis an agglomerated finite element(AFE)basis for the DG method on general polygons,since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles.The basis functions on each sub-triangle are defined,as usual,on a universal reference element,hence allowing to compute universal mass,flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only.Consequently,the construction of an efficient quadrature-free algorithm is possible,despite the unstructured nature of the computational grid.High order of accuracy in time is achieved thanks to the ADER approach,making use of an element-local space-time Galerkin finite element predictor.The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations.The numerical results have been checked with reference solutions available in literature and also systematically compared,in terms of computational efficiency and accuracy,with those obtained by the corresponding modal DG version of the scheme.
