Dual-consistent approximations of PDEs and super-convergent functional output
نویسندگان
چکیده
Finite-difference operators satisfying the summation-by-parts (SBP) rule can be used to obtain high-order accurate time-stable schemes for partial differential equations (PDEs), when the boundary and interface conditions are imposed weakly by the simultaneous approximation term method (SAT). The most common SBP-SAT discretizations are accurate of order 2p in the interior and order p close to the boundaries, which yields a global accuracy of p+ 1 for first order PDEs. However, Berg and Nordström [2012] and Hicken and Zingg [2014] have shown that any linear functional computed from the time-dependent numerical solution will be accurate of order 2p if the spatial discretization is dual-consistent. This text discusses the concepts of dual problem and dual consistency. We walk through some of the definitions and results in Berg and Nordström [2012] and Hicken and Zingg [2014] while emphasizing the underlying functional analysis considerations. 1 Dual problems and dual consistency We introduce the concepts of dual problem and dual consistency by considering linear partial differential equations with homogeneous boundary conditions. Let L be a linear differential operator of order m on a domain Ω and consider the problem Lu− f = 0, x ∈ Ω, (1.1) ∗Division of Scientific Computing, Department of Information Technology, Uppsala university, SE-751 05 Uppsala, Sweden. [email protected]
منابع مشابه
A Hybrid Fourier-Chebyshev Method for Partial Differential Equations
We propose a pseudospectral hybrid algorithm to approximate the solution of partial differential equations (PDEs) with non-periodic boundary conditions. Most of the approximations are computed using Fourier expansions that can be efficiently obtained by fast Fourier transforms. To avoid the Gibbs phenomenon, super-Gaussian window functions are used in physical space. Near the boundaries, we use...
متن کاملUsing Chebyshev polynomial’s zeros as point grid for numerical solution of nonlinear PDEs by differential quadrature- based radial basis functions
Radial Basis Functions (RBFs) have been found to be widely successful for the interpolation of scattered data over the last several decades. The numerical solution of nonlinear Partial Differential Equations (PDEs) plays a prominent role in numerical weather forecasting, and many other areas of physics, engineering, and biology. In this paper, Differential Quadrature (DQ) method- based RBFs are...
متن کاملNew approximations for the cone of copositive matrices and its dual
We provide convergent hierarchies for the convex cone C of copositive matrices and its dual C∗, the cone of completely positive matrices. In both cases the corresponding hierarchy consists of nested spectrahedra and provide outer (resp. inner) approximations for C (resp. for its dual C∗), thus complementing previous inner (resp. outer) approximations for C (for C∗). In particular, both inner an...
متن کاملDual Z-Source Network Dual-Input Dual-Output Inverter
This paper presents a modified nine switch inverter with two inputs and two Z-source networks. This inverter has two DC inputs and two AC outputs. Input DC voltages can be boosted to the required level. Amplitude, frequency and phase of AC output voltages can be controlled, independently. The proposed converter can be used in applications with two unregulated DC sources, which require feeding ...
متن کاملGeneralized Kernel Classification and Regression
We discuss kernel-based classification and regression in a general context, emphasizing the role of convex duality in the problem formulation. We give conditions for the existence of the dual problem, and derive general globally convergent classification and regression algorithms for solving the true (i.e. hard-margin or rigorous) dual problem without resorting to approximations.
متن کامل