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We consider finite difference approximations of the second derivative, exemplified in Poissons equation, the heat equation and the wave equation. The finite difference operators satisfy a summation-by-parts property, which mimics the integration-by-parts. Since the operators approximate the second derivative, they are singular by construction. To impose boundary conditions, these operators are modified using Simultaneous Approximation Terms. This makes the modified matrices non-singular, for most choices of boundary conditions. Recently, inverses of such matrices were derived. However, when considering Neumann boundary conditions on both boundaries, the modified matrix is still singular. For such matrices, we have derived an explicit expression for the Moore-Penrose pseudoinverse, which can be used for solving elliptic problems and some time-dependent problems. The condition for this new pseudoinverse to be valid, is that the modified matrix does not have more than one zero eigenvalue. We have reconstructed the sixth order accurate narrow-stencil operator with a free parameter and show that more than one zero eigenvalue can occur. We have performed a detailed analysis on the free parameter to improve the properties of the second derivative operator. We complement the derivations by numerical experiments to demonstrate the improvements of the new second derivative operator.
In this work, new finite difference schemes are presented for dealing with the upper-convected time derivative in the context of the generalized Lie derivative. The upper-convected time derivative, which is usually encountered in the constitutive equ
We develop an energy-based finite difference method for the wave equation in second order form. The spatial discretization satisfies a summation-by-parts (SBP) property. With boundary conditions and material interface conditions imposed weakly by the
We present a hybridization technique for summation-by-parts finite difference methods with weak enforcement of interface and boundary conditions for second order, linear elliptic partial differential equations. The method is based on techniques from
A second-order accurate in time, positivity-preserving, and unconditionally energy stable operator splitting numerical scheme is proposed and analyzed for the system of reaction-diffusion equations with detailed balance. The scheme is designed based
This paper extends the second-order accurate BGK finite volume schemes for the ultra-relativistic flow simulations [5] to the 1D and 2D special relativistic hydrodynamics with the Synge equation of state. It is shown that such 2D schemes are very tim