DSS Library Routines The source code for the dss routines (differentiation in space subroutines) in Fortran, Matlab and R is in this download site. Some details concerning these routines are provided next: Fortran The dss routines were first written in Fortran and include the following: First derivative: dss002,dss004,dss006,dss008,dss010 Second derivative: dss042,dss044,dss046,dss048,dss050 Hyperbolic (convective) PDEs: dss012 - two-point upwind finite differences (FDs) dss018 - four-point biased upwind FDs dss020 - five-point biased upwind FDs Single and double precision routines have the extensions .fsp and .fdp, respectively. When these routines were written, upwind FDs were used to approximate hyperbolic PDEs. The upwind approximations were generally satisfactory if the PDE solutions did not have steep moving fronts or discontinuities. Additional details are given in ref. (1). (1) Schiesser, W. E. (1991), The Numerical Method of Lines Integration of Partial Differential Equations, Academic Press, San Diego, pp 128-141, 211-213 Matlab First derivative: dss002,dss004,dss006,dss008,dss010 Second derivative: dss042,dss044,dss046,dss048,dss050 Hyperbolic (convective) PDEs: dss012 - two-point upwind finite differences (FDs) dss020 - five-point biased upwind FDs Flux limiters: smart.m, super.m, vanl.m for the smart, superbee, vanLeer limiters (which use max3.m) The flux limiters provide better resolution of hyperbolic PDE solutions that exhibit steep moving fronts or discontinuites than FDs. Additonal details are given in refs. (7), (8). (7) Griffiths, G. W., and W. E. Schiesser (2012), Traveling Wave Analysis of Partial Differential Equations: Numerical and Analytical Solutions with Matlab and Maple, Academic Press/Elsevier, Burlington, MA, pp 37-43 (8) Schiesser, W. E. (2013), Partial Differential Equation Analysis in Biomedical Engineering, Cambridge University Press, Cambridge, UK, Chapter 1 R First derivative: dss002,dss004,dss006,dss008,dss010 Second derivative: dss042,dss044,dss046,dss048,dss050 Hyperbolic (convective) PDEs: dss012 - two-point upwind finite differences (FDs) dss020 - five-point biased upwind FDs Flux limiters: smart.R, super.R, vanl.R for the smart, superbee, vanLeer limiters (which use max3.R) The flux limiters provide better resolution of hyperbolic PDE solutions that exhibit steep moving fronts or discontinuites than FDs. Additonal details are given in refs. (7), (8). (7) Griffiths, G. W., and W. E. Schiesser (2012), Traveling Wave Analysis of Partial Differential Equations: Numerical and Analytical Solutions with Matlab and Maple, Academic Press/Elsevier, Burlington, MA, pp 37-43 (8) Schiesser, W. E. (2013), Partial Differential Equation Analysis in Biomedical Engineering, Cambridge University Press, Cambridge, UK, Chapter 1 Refs. (7), (8) pertain to the use of flux limiters in Matlab. However, the limiters in R are used in exactly the same way. The dss and flux limiter routines are for 1D PDEs, but they can also be applied to 2D and 3D PDEs with some additional programming. This procedure is explained through examples in ref. (6). (6) Schiesser, W. E., and G. W. Griffiths (2009), A Compendium of Partial Differential Equations Models: Method of Lines Analysis with Matlab, Cambridge University Press, Cambridge, UK, 2D: Chapters 10,13,14, 3D: Chapter 11