## An Iterative Method for the Solution of Sets of First-Order Hyperbolic Differential Equations

January 15th, 1967

### Paper in the SIAM Journal “Applied Math”, volume 15, no 1, January 1967.

*S F Bush.*

#### Introduction

The increasing sophistication of many processes in the chemical and mechanical engineering industries usually demands some mathematical expression of the interaction of the physical and chemical quantitites involved. Frequently attempts are made to predict the behaviour of a plant by writing equations based on the conservation of matter, energy and momentum in the system.

Often such mathematical models are composed of one or more hyperbolic partial differential equations and, if the transient behaviour is required, these equations have to be solved for step changes in the dependent variables at the boundaries. There is a well-known difficulty associated with the numerical solution of hyperbolic equations and that is the propagation of discontinuities in the dependent variables. In compressible flow, discontinuities appear as shocks, and in order to render the equations susceptible to accurate numerical analysis, von Neumann and Richtmyer[1] proposed the introduction of a pseudo-viscosity term which has the effect of smoothing the discontinuity and making the solutions continuous. Similar devices have been suggested in connection with the solution of hyperbolic equations arising from mass and heat transfer systems (see, for example, [2], but in general they have not been pursued owing to complications in the specification of boundary conditions.

While a number of solutions are available for relatively simple physical systems, for example, the pebble-bed regenerator,[3] and a linearized model of the packed distillation-column[4], considerable difficulty has been found in solving two or more simultaneous hyperbolic equations. Much of the engineering literature on the transient behaviour of distributed flow-systems to 1961 is reviewed by Rosenbrock[5], Williams and Morris[6] (distillation columns), and Archer and Rothfus[7](heat exchangers). This material is concerned with both numerical and approximate analytical solutions, particularly for control studies, although not all the mathematical models are based on sets of hyperbolic differential equations. More recently, Stone and Brian[8] proposed a general finite difference equation for application to convective transport problems, which are often represented by hyperbolic equations, but their analysis also demonstrated implicitly that if accurate representation of discontinuities is required, the equations must be integrated along the characteristic curves of the system. In this paper we deal with systems where discontinuities constitute an important feature of the solution, and a means of finding both accurate numerical solutions and analytic approximations is proposed.

#### References

[1] J von Neumann and R D Richtmyer, *A method for the numerical calculation of hydrodynamic shocks*, J. Appl. Phys., 21 (1950), pp 232-237.

[2] J R Bowman and R C Briant, *Theory of the performance of packed rectifying columns*, Ind. Eng. Chem., 39 (1947), pp 745-751.

[3] T E W Schumann, *Heat transfer: a liquid flowing through a porous prism*, J. Franklin Inst., 208 (1929), pp 405-416.

[4] K Cohen, *Packed fractionating columns and the concentration of isotopes*, J. Chem. Phys., 8 (1940), pp 588-597.

[5] H H Rosenbrock, *Transient processes in distillation columns for separating multicomponent mixtures*, Trans. Inst. Chem. Engrs., 40 (1962), pp 376-384.

[6] T J Williams and H J Morris, *A survey of the literature on heat-exchanger dynamics and control*, Chemical Engineering Progress Symposium Series, no 36, vol 57, American Institute of Chemical Engineers, (1961) pp 20-33.

[7] D H Archer and R R Rothfus, *Dynamics and control of distillation units and other mass transfer equipment*, Chemical Engineering Progress Symposium Series, no 36, vol 57, American Institute of Chemical Engineers, (1961), pp 2-19.

[8] H L Stone and P L T Brian, *Numerical solution of convective transport problems*, A.I.Ch.E.J., 9 (1963), pp 681-688.

[9] S L Sobolev, *Equations of Mathematical Physics*, Pergamon Press, Oxford, (1964), Lecture 22.

[10] I G Petrovsky, *Lectures on Partial Differential Equations*, A. Shenitzer, transl., Interscience, New York, (1954), p 65.

[11] S F Bush, Doctoral thesis, University of Cambridge, (1965).

[12] _____, *Approximate solutions of the equations of flow systems*, to appear.

[13] R Courant and D Hilbert, *Methods of Mathematical Physics*, vol II, Interscience, New York (1962).