Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Optimum positive linear schemes for advection in two and three dimensions
SIAM Journal on Numerical Analysis
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
Toward the ultimate conservative scheme: following the quest
Journal of Computational Physics
A residual-based compact scheme for the compressible Navier-Stokes equations
Journal of Computational Physics
Residual Distribution Schemes for Conservation Laws via Adaptive Quadrature
SIAM Journal on Scientific Computing
High Order Fluctuation Schemes on Triangular Meshes
Journal of Scientific Computing
Journal of Computational Physics
Anti-diffusive flux corrections for high order finite difference WENO schemes
Journal of Computational Physics
Journal of Computational Physics
Essentially non-oscillatory Residual Distribution schemes for hyperbolic problems
Journal of Computational Physics
Residual Distribution Schemes on Quadrilateral Meshes
Journal of Scientific Computing
A first-order system approach for diffusion equation. II: Unification of advection and diffusion
Journal of Computational Physics
High Order Finite Difference WENO Schemes for Nonlinear Degenerate Parabolic Equations
SIAM Journal on Scientific Computing
Visualization of Advection-Diffusion in Unsteady Fluid Flow
Computer Graphics Forum
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In this paper, we propose a high order residual distribution conservative finite difference scheme for solving convection-diffusion equations on non-smooth Cartesian meshes. WENO (weighted essentially non-oscillatory) integration and linear interpolation for the derivatives are used to compute the numerical fluxes based on the point values of the solution. The objective is to obtain a high order scheme which, for two space dimension, has a computational cost comparable to that of a high order WENO finite difference scheme and is therefore much lower than that of a high order WENO finite volume scheme, yet it does not have the restriction on mesh smoothness of the traditional high order conservative finite difference schemes, hence it would be more flexible for the resolution of sharp layers. The principles of residual distribution schemes are adopted to obtain steady state solutions. The distribution of residuals resulted from the convective and diffusive parts of the PDE is carefully designed to maintain the high order accuracy. The proof of a Lax-Wendroff type theorem is provided for convergence towards weak solutions in one and two dimensions under additional assumptions. Extensive numerical experiments for one and two-dimensional scalar problems and systems confirm the high order accuracy and good quality of our scheme to resolve the inner or boundary layers.