Matrix analysis
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
The stability of numerical boundary treatments for compact high-order finite-difference schemes
Journal of Computational Physics
Multigrid strategies for viscous flow solvers on anisotropic unstructured meshes
Journal of Computational Physics
A stable and conservative interface treatment of arbitrary spatial accuracy
Journal of Computational Physics
Journal of Computational Physics
High-order finite difference methods, multidimensional linear problems, and curvilinear coordinates
Journal of Computational Physics
Finite volume methods, unstructured meshes and strict stability for hyperbolic problems
Applied Numerical Mathematics
A UNIFIED MULTIGRID SOLVER FOR THE NAVIER-STOKES EQUATIONS ON MIXED ELEMENT MESHES
A UNIFIED MULTIGRID SOLVER FOR THE NAVIER-STOKES EQUATIONS ON MIXED ELEMENT MESHES
A stable hybrid method for hyperbolic problems
Journal of Computational Physics
Stable artificial dissipation operators for finite volume schemes on unstructured grids
Applied Numerical Mathematics
A stable and efficient hybrid scheme for viscous problems in complex geometries
Journal of Computational Physics
An accuracy evaluation of unstructured node-centred finite volume methods
Applied Numerical Mathematics
Analysis of the order of accuracy for node-centered finite volume schemes
Applied Numerical Mathematics
An adaptive implicit-explicit scheme for the DNS and LES of compressible flows on unstructured grids
Journal of Computational Physics
Journal of Computational Physics
Optimal diagonal-norm SBP operators
Journal of Computational Physics
A generalized framework for nodal first derivative summation-by-parts operators
Journal of Computational Physics
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Our objective is to analyse a commonly used edge based finite volume approximation of the Laplacian and construct an accurate and stable way to implement boundary conditions for time dependent problems. of particular interest are unstructured grids where the strength of the finite volume method is fully utilised. As a model problem we consider the heat equation. We analyse the Cauchy problem in one and several space dimensions and prove stability on unstructured grids. Next, the initial-boundary value problem is considered and a scheme is constructed in a summation-by-parts framework. The boundary conditions are imposed in a stable and accurate manner, using a penalty formulation. Numerical computations of the wave equation in two-dimensions are performed, verifying stability and order of accuracy for structured grids. However, the results are not satisfying for unstructured grids. Further investigation reveals that the approximation is not consistent for general unstructured grids. However, grids consisting of equilateral polygons recover the convergence.