A fast stable discretization of the Constant–Convection–Diffusion–Reaction equations of Kinetic Capillary Electrophoresis (KCE)

Abstract

A discretization scheme is introduced for a set of convection-diffusion equations with a non-linear reaction term, where the convection velocity is constant for each reactant. This constancy allows a transformation to new spatial variables, which ensures the global stability of discretization. Convection-diffusion equations are notorious for their lack of stability, arising from the algebraic interaction of the convection and diffusion terms. Unexpectedly, our implemented numerical algorithm proves to be faster than computing exact solutions derived for a special case, while remaining reasonably accurate, as demonstrated in our runtime and error analysis.MSC class: 65M08 (primary); 65M12, 35C05 (secondary).

A Computational Resolution of the Inverse Problem of Kinetic Capillary Electrophoresis (KCE)

“…Thus the runtime of the inverse solver is fundamentally implied by that of the direct solver. The stability and efficiency of the direct solver was established in our earlier article [12].…”

“…The direct problem of efficiently generating accurate numerical solutions to the set of partial differential equations of Kinetic Capillary Electrophoresis (KCE) [6], has been resolved earlier via a multimesh algorithm [12], which fully overcomes the typical instability arising from the interaction of the diffusion and convection terms.…”

“…We adopt the physical model and the notations covered in earlier articles [12,13], originally introduced in [6,3,9].…”

“…The direct problem of efficiently generating accurate numerical solutions to the set of partial differential equations of Kinetic Capillary Electrophoresis (KCE) [6], has been resolved earlier via a multimesh algorithm [12], which fully overcomes the typical instability arising from the interaction of the diffusion and convection terms. This potent solution to the direct problem allows the effective resolution of the inverse problem on a reasonable timescale, as we shall hereby discuss and demonstrate.…”

“…See our previous articles for further details [13,12,6], such as the physical meaning of the above constants.…”