 In this paper, we elaborate the analysis of some of the schemes which were presented in arXiv:1711.07926 for the heat equation with periodic boundary conditions. We adopt this methodology to derive finite-difference schemes for heat equation with Dirichlet and Neumann boundary conditions, whose convergence rates are higher than their truncation errors. We call these schemes error inhibiting schemes. When constructing a semi-discrete approximation to a partial differential equation (PDE), a discretization of the spatial operator has to be derived. For stability , this discrete operator must be semi-bounded. Under this semi-boundness, the Lax-Richtmyer equivalence theorem assures that the scheme converges at most, as the order of the error will be at most of the order of the truncation error. Usually, the error is in indeed of the order of the truncation error. In this paper, we demonstrate that schemes can be constructed such that their errors are smaller than their truncation errors. This property can enable us design schemes which are more efficient than standard schemes.
Finite Difference (FD) schemes are widely used in science and engineering for approximating solutions of partial differential equations (PDEs). Error analysis of FD schemes relies on estimating the truncation error at each time step. This approach usually leads to a global error whose order is of the same order of the truncation error. For classical FD schemes the global error is indeed of the same order as the truncation error. A particular class of FD schemes is the Block Finite Difference (BFD) schemes, in which the grid is divided into blocks. The structure of such schemes is similar to the structure of the Discontinuous Galerkin (DG) method, and allows inhabitation of the truncation errors. Recently, much effort was devoted to design BFD schemes whose global error converges faster than the truncation error. In this paper, we elaborate the approach presented in arXiv:1711.07926 for the heat equation with periodic boundary conditions. We generalize this methodology to design BFD schemes for the heat equation with Dirichlet or Neumann boundary conditions, whose global error converges faster than the truncation error. Such schemes are henceforth called Error Inhibiting Schemes. We provide an explicit error analysis, including proofs of stability and convergence of the proposed schemes. We illustrate our approach using several numerical examples, which demonstrate the efficiency of our method in comparison to standard FD schemes.
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Finite Difference (FD) schemes are widely used in science and engineering for approximating solutions of partial differential equations (PDEs). Error analysis of FD schemes relies on estimating the truncation error at each time step. This approach usually leads to a global error whose order is of the same order of the truncation error. For classical FD schemes the global error is indeed of the same order as the truncation error. A particular class of FD schemes is the Block Finite Difference (BFD) schemes, in which the grid is divided into blocks. The structure of such schemes is similar to the structure of the Discontinuous Galerkin (DG) method, and allows inhabitation of the truncation errors. Recently, much effort was devoted to design BFD schemes whose global error converges faster than the truncation error.
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convergence rates are higher than their truncation errors. We call these schemes error inhibiting schemes. When constructing a semi-discrete approximation to a partial differential equation (PDE), a discretization of the spatial operator has to be derived. For stability , this discrete operator must be semi-bounded. Under this semi-boundness, the Lax-Richtmyer equivalence theorem assures that the scheme converges at most, as the order of the error will be at most of the order of the truncation error. Usually, the error is in indeed of the order of the truncation error. In this paper, we demonstrate that schemes can be constructed such that their errors are smaller than their truncation errors. This property can enable us design schemes which are more efficient than standard
global error converges faster than the truncation error. Such schemes are henceforth called Error Inhibiting Schemes. We provide an explicit error analysis, including proofs of stability and convergence of the proposed schemes. We illustrate our approach using several numerical examples, which demonstrate the efficiency of our method in comparison to standard FD