For a given model scenario with initial values and prescribed emission and deposition rates (i.e., constant sources and sinks for gas phase species), the user is able to simulate the evolution of species concentrations and other variables in a box-model framework over a defined time interval. The Cminor box-model is depicted in Fig. .

The syntax rules for combustion mechanisms can be taken from the ChemKin documentation , which is widely used for combustion kinetics. The necessary thermodynamic data base, which includes the coefficients for fitting polynomials, follows the same ChemKin syntax rules . For atmospheric chemical mechanisms, a detailed description of the syntax rules is provided in the Supplement. In any case, the input data has to be provided in readable ASCII format. Assistance in the syntax rules is also given by the examples provided in the repository, as well as on request by the authors. A short summary shall be given in the following.

Running a simulation requires four files beside the actual model code:

the *.sys-file,

The liquid water content (LWC) describes the mass of condensed water per volume of air. The LWC for a mono-dispersed droplet population is given by LWC =ρwVN, where ρw=1000 kg m-3 is the mass density of liquid water, V the droplet volume, and N the number concentration of droplets. Within this simplified model, the time dependent LWC(t) is given by a piecewise linear function mimicking the condensation process, where t is the independent time variable, b1,…,b6 are the bounds of the linear function, and LWC_min⁡ and LWC_max⁡ the respective minimum and maximum LWC. If the simulation time exceeds the bound b6, the function will be continued periodically, such that bi′=bi+(b6-b1) for i=1,…,6, as indicated in Fig. . From LWC(t), the droplet wet radii are calculated and used for aqueous chemistry and phase transfer. In the case of an adiabatic parcel, the LWC is determined by the droplet condensation equation, which will be described in Sect. , below. For prescribed LWC(t), all droplets contain the same amount of liquid water while different amounts of dissolved aerosol are possible.

The composition of aerosol is initialized at the beginning of the simulation. The syntax for the setup of the aerosol modes is described in the Supplement, Sect. S6.3. We assume that the aerosol is completely dissolved in the droplets. The aerosol is assumed to be log-normally distributed with an arbitrary number of modes. Each mode can have a specific composition, resembling, e.g., coexisting sea-salt and ammonium-salt particles, or any internally and/or externally mixed aerosol distribution. The aerosol is assigned to a user-specified number of droplet classes, each representing a number of equal hydrometeors with a specific amount of liquid water and dissolved aerosol mass. The aerosol mass is determined by discretizing the aerosol distribution using logarithmically equidistant bins, one for each droplet class. To constrain the aerosol masses to a reasonable range, the aerosol distribution of each mode is bounded by determining the 0.5 %-radius-quantile and the 99.5 %-mass-quantile. If all modes contain the same aerosol type, this procedure is applied for the sum of these modes, using the respective smallest and largest quantiles of the individual modes. If the modes differ in aerosol type, the procedure is applied to each mode individually. In addition to the initial aerosol mass, general initial values for any aqueous species can be prescribed.

If the adiabatic parcel option is used, aerosol activation is determined by Köhler theory (Sect. ). Alternatively, when the change in LWC is prescribed (Sect. ), only aerosols with dry radii greater than a user-specified threshold are assumed to be activated, and the LWC is distributed equally among the activated particles.

Consider a system of nR reactions involving nS chemical species that can be represented in the general form 1∑j=1nSνijeSj→ki∑j=1nSνijpSj,i=1,…,nR, where the stoichiometric coefficients of educts νije and products νijp are stored as sparse matrices. Sj denotes the name of the jth chemical species in the mechanism and ki is the rate constant. Table  summarizes the submodules that provide the rate constant ki for the case of atmospheric chemical mechanisms, a detailed description of all implemented types of rate constants can be seen in the Supplement, Sect. S4.

All functions are implemented in a way that minimizes numerical operations, such as (+,-,*,/,**), and calls to intrinsic functions, e.g., exp⁡(⋅), log⁡(⋅), sin⁡(⋅) and cos⁡(⋅), while performing these efficiently. Next to hard coded functions, the user is also able to declare own functions that do not match any of the functions provided, see Supplement, Sect. S4.1, to introduce custom rate constants.

The general reaction kinetics formulation Eq. () leads to a set of nS ODEs expressing mass conservation in the system as 2dcjdt=∑i=1nRνijri(c,T,ϕ)+cjemis,j=1,…,nS where dcj/dt is the change in concentration of species j, νij=(νijp-νije) is the difference between stoichiometric coefficients of products and educts of species j in reaction i, and ri(c,T,ϕ) is the reaction rate, depending on the concentrations of all reactants in reaction i, expressed by c, temperature T, and possibly other parameters ϕ. In addition, one can specify emission and deposition rates for gas phase species, which are summed up in the vector cemis. While the emission rates are specified as actual rate in the *.ini-file, deposition is determined by a given rate constant from the *.ini-file. The sink/source term cjemis is calculated as cjemis=cjin-cjoutcj,j=1,…,nS, where cjin is the given emission rate and cjout the given deposition rate constant. The reaction rate is calculated by 3ri(c,T,ϕ)=ki(T,ϕ)⋅∏j=1nScjνije,i=1,…,nR. The rate constants ki may depend on temperature and other parameters like insolation (introducing a time dependency), or third body collisions (concentration dependency). The temperature plays a major role in combustion, and a respective equation is included in the calculations, as described later. While rate constants with non-temperature dependencies are implemented and can be used, they are not treated as non-autonomous parts of the system in the numerical procedure, beside being evaluated whenever needed. This is a common approximation and assumes not too rapid changes (e.g., a constant solar zenith angle, which is a good approximation for typical time steps below a few minutes, or third body collisions, that depend on species that are basically passive).

The right-hand side of Eq. () represents the law of mass action. In general, several other concentrations influence the production rate of each species, therefore the ODE system is highly coupled.

Because Cminor is capable of simulating gas phase combustion mechanisms, the initial value problem needs closure through energy conservation. A perfectly adiabatic constant-volume reactor is considered, as this condition is the one usually occurring when incorporating the solution of detailed chemistry into multidimensional computational fluid dynamics (CFD) codes, where the species and internal energy source terms for each cell are computed as part of a sub-cycling strategy for the whole code . In particular, the internal energy source term gives a change in the average reactor temperature 4dTdt=-1c¯v(c,T)ρ∑j=1nS∑i=1nRUj(T)[νijri(c,T,ϕ)+cjemis], containing the temperature dependent polynomial fitting function Uj(T) describing the molar internal energy values of species j , the mass-averaged specific heat at constant volume c¯v(c,T)=∑j=1nS(cj∂Uj(T)/∂T), and the average system density ρ.

The gaseous-aqueous mass transfer in multi-phase systems has to be defined by a pseudo first-order equilibrium reaction A(g)⇌A(a), where subscript g denotes the concentration of species A in the gas phase and subscript a the concentration of A in the aqueous phase, respectively. The forward and backward rate coefficients are calculated using the accommodation coefficient α, the gas phase diffusion constant Dg, and Henry's law constant. The correct syntax for species names is denoted in the Supplement, Sect. S3.3.

With Henry's law coefficient A:=KH⊝ in [molL-1atm-1] and B:=-△solnH/R in [K], where △solnH= molar enthalpy of dissolution in [Jmol-1] and R is the gas constant. Then, Henry's law constant kH is computed using the temperature-dependent equation 5kH(T)=A⋅exp⁡B⋅T-1-Tref-1, where A and B are the parameter given in the third line of the reaction structure (Table ), T is the actual temperature, and Tref=298.15 K is a reference temperature. The droplet wet radius r is then used to calculate the mass transfer coefficient according to , 6kmt=kdiff⋅r2+kacc⋅r-1, with kdiff=1/(3Dg), where Dg is the gas phase diffusion coefficient, and kacc=4/(3α^v), where α^ is the mass accommodation coefficient, v=8RT/(πmmol) is the mean molecular speed, and mmol the molar mass of the gas. The reaction rates for gas uptake by droplets is 7dA(a)dt|gas uptake=-dA(g)dt|gas uptake=kmtLWCiρw[A(g)], and 8dA(g)dt|gas escape=-dA(a)dt|gas escape=kmt[A(a)]/(kH(T)RatmT) for the escape of gas from droplets, with Ratm=0.0820574 Latmmol-1K-1. The division by ρw converts the LWC from [kgm-3] to [m3m-3], ensuring unit independent rates. The underscore i indicates that if there are multiple droplet classes, only the water volume of a single droplet class is determining the uptake of gases by this class.

In terms of the liquid water mixing ratio of droplet class i, ql,i, the equation for condensational growth of water mass reads 9dql,idt=ni4πrd,i2S-Seq,iFkSeq,i(rd,i+rβ)+Fd(rd,i+rα), where rd,i is the radius of the droplets of this class and ni the corresponding number mixing ratio of droplets. S=qvqS(T) is the ambient relative humidity with qv and qS(p,T) being current and saturation vapor mixing ratio, respectively. 10Seq,i=nw,inw,i+ns,iexp⁡2σrd,iρwRvT is the value of the Köhler curve for the particles' current radius, where nw,i and ns,i are the concentrations of liquid water and dissolved species in moles, respectively, σ is the surface tension between liquid water and air, and Rv is the gas constant of water vapor. 11Fk=LvkTTLvRvT-1andFd=RvTDves(T) are variables accounting for latent heat release and diffusion of vapor . Lastly, 12rβ=kTβH2Op(2πRaT)12cva+Ra/2andrα=2πRvT12DvαH2O are physical relaxation parameters . To follow current common nomenclature, we changed the names from the original reference, because they refer to what is now called “accommodation coefficient“ as “condensation coefficient“. We kept the letter α referring to what is now usually called the accommodation coefficient, as for the gases in Sect. . By recommendations in and , we set βH2O=1.0 and αH2O=0.0415. These values can be changed to the user's preferences without recompilation, as described in the Supplement, Sect. S9.4.

We have to include a few other equations to account for an adiabatically rising parcel. The water vapor mixing ratio qv is only affected by condensation and evaporation, i.e., the change in liquid water mixing ratio ql, such that dqvdt=-dqldt with ql=∑i=1nDql,i, where nD is the number of droplet classes. A user-specified updraft velocity dzdt=w determines the rate of pressure change, where we assume instantaneous adjustment of the parcel's pressure to the ambient conditions. From the ideal gas law, 13ρa=pRaT, where ρa is the mass density of the air in the parcel and Ra the specific gas constant of dry air. Differentiation in time gives 14dρadt=-ρagRaTdzdt-pRaT2dTdt. Here, dT/dt considers the expanding parcel (dry adiabatic lapse rate) and latent heat release following Eq. (), such that 15dTdt=-gcpadzdt+Lvcpadqldt.

This section will give a brief introduction to Rosenbrock methods in order to clarify the notation. The starting point is the autonomous initial value problem 16y˙=f(y),t>t0,y(t0)=(y1,0,…,yn,0)T. This assumption does not restrict the problem for non-autonomous systems y˙=f(t,y), because it can easily be transformed by treating the time variable t as dependent, such that t˙=1. The ODE system contains only first order derivatives dy/dt, which are usually non-linear functions f of concentration. Most stiff solvers benefit from using implicit formulations such as Rosenbrock methods which arise from diagonal-implicit Runge-Kutta (DIRK) methods. The simplest scheme of this kind is the backward Euler method 17yn+1=yn+hfyn+1, where h is the step size and yn the current state vector. To avoid implicit systems and iterative solvers, Eq. () can be linearized, leading to 18yn+1=yn+hfyn+hyn+1-ynf′yn. This is commonly referred to as a linear-implicit method. It can be seen as restricting the iterative solver to one step of Newton's method, see for elaborate derivations. To achieve a higher order of consistency, proposed to generalize this linearly implicit approach to methods using more stages. The general form of a non-autonomous s-stage Rosenbrock method produces a next step solution as 19yn+1=yn+∑i=1sbik(i),withk(i)=hftn+αih,yn+∑j=1i-1αijk(j)+γih2fttn,yn+hJtn,yn⋅∑j=1iγijk(j), where the coefficients bi, αij, and γij define the particular method, and are chosen such that a certain accuracy and stability is granted . J refers to the Jacobian of the system. For linear-implicit methods the method coefficients have the additional feature that αij=0 for all j≥i, γij=0 for all j≥i+1, and 20αi=∑j=1i-1αij,γi=∑j=1iγij. The partial derivative ft in Eq. () is equal to zero for autonomous systems. Reducing the computation cost is crucial for large systems because each time step requires not only s-1 matrix-vector products, but also s matrix decompositions. It is therefore advantageous to choose the same diagonal elements for γii=γ, i=1,…,s. Thus, only one decomposition is needed per time step. In order to avoid the matrix-vector products on the right-hand side of Eq. (), the stage vectors k(i) are substituted by u(i)=∑jiγijk(j) for i=1,…,s see. The modified Rosenbrock method with s internal stages is then given by 21yn+1=yn+∑j=1smju(j),with(I-hγJ)u(i)=hf(tn+αih,yn+∑j=1i-1aiju(j))+γih2ft(tn,yn)+∑j=1i-1diju(j),i=1,…,s, where I is the identity matrix, and Γ=(γii), (aij)=(αij)Γ-1, (dij)=diag(γ11-1,…,γss-1)-Γ-1, (mj)=(bj)Γ-1, and γ the method parameters.

Usually, the purpose of adaptive step size control is to achieve some predetermined accuracy in the solution with minimum computational effort. The use of Rosenbrock methods allows us to calculate the local error of the current state vector y. This task can be carried out by computing a second state vector y^, which is given by an embedded formula where the order of the embedded method p^ is usually lower than order p of the actual formula (p^=p-1). The step size control is implemented by estimating the local error e^j=yj-y^j, and is then scaled by the denominator scalj=toljA+max⁡{|yj|,|y^j|}⋅tolR, where toljA is the absolute and tolR the relative tolerance given by the user. Note that toljA can be set for each species separately, where tolR is equal for all species. Cminor's implementation allows to specify two different absolute tolerances, for gaseous and aqueous species. The actual measurement of the error induced by the Rosenbrock method is computed either via the maximum norm or Euclidean norm, denoted by ∥⋅∥. The new step size is then calculated by the following expression: 22hnew=max⁡{hmin⁡,holdmax⁡{0.1,0.8∥e^∥-δ}}, if ∥e^∥>1,min⁡{hmax⁡,0.8hold∥e^∥-δ,2hold}, otherwise.  The exponent δ is defined as δ=1/(p+1). A new step size hnew is estimated that will yield an error of tolR on the next step or the next try at taking this step, as the case may be. The step size is multiplied by 0.8 to avoid failures. In addition, the step size hnew is bounded by user-given values for a minimal and a maximal step size, hmin⁡ and hmax⁡, respectively.

The initial value problem is described by Eqs. () and (), where the dimension of the ODE system is n=nS+1, with c˙∈RnS and T˙∈R. The ODE system can then be represented in a very short manner as matrix-vector products, 23y′=f(c,T)=c˙T˙=νTr+cemis-1c¯vρUT[νTr+cemis],y(t0)=(c10,…,cnS0,T0)T. The use of Rosenbrock methods Eq. () requires the calculation of the Jacobian matrix of the ODE system Eq. (), 24J=∂f(c,T)∂(c,T)=⋱⋰⋮∂c˙l∂cj∂c˙l∂T⋰⋱⋮⋯∂T˙∂cj⋯∂T˙∂T,j=1,…,nSl=1,…,nS, where the first block ∂c˙l/∂cj is a square sparse matrix of dimension nS containing the derivatives of the change of cl with respect to cj. The simplification of this part of the Jacobian matrix is related to the assumption that k=k(T) in Eq. () depends only on temperature and not on other variables, e.g., time, pressure, or concentrations of catalytic species. This simplification is justified by the assumed slow change of these parameters, such that 25Jcc=∂∂cjdcldt=∑i=1nR[νil∂ri∂cj]=νTDrνeDc-1,j=1,…,nS,l=1,…,nS, where Dr=diag(r1,…,rnR) and Dc-1=diag(1/c1,…,1/cnS) are diagonal matrices containing the reactions rates and the inverse concentrations, respectively. The second block ∂c˙l/∂T consists of a full column vector and represents the derivatives of change in cl with respect to T. These are expressed as 26JcT=∂∂Tdcldt=∑i=1nR[νil∂ri∂T]=νTDrK,l=1,…,nS, where Ki=∂ki/∂T⋅ki-1 contains the derivatives of the ith rate constants with respect to T, multiplied by the inverse rate constant of reaction i. As mentioned before, ki is assumed to be only temperature-dependent. The third part of the Jacobian matrix ∂T˙/∂cj is a full row vector and contains the derivatives of change in T with respect to cj. It is expressed as 27JTc=∂∂cjdTdt=-1c¯vρ[Cv,jT˙+∑l=1nSUj∂c˙l∂cj],j=1,…,nS=-1c¯vρ[CvT˙+UTνTDrνeDc-1], where Cv,j=∂Uj/∂T is the constant volume specific heat of the jth species. The first item in the Jacobian matrix, i.e., the entry in position (nS+1,nS+1), is the derivative of the average mixture temperature change rate with respect to the system temperature itself, 28JTT=∂∂TdTdt=-1c¯vρ{ρT˙∂c¯v∂T+∑l=1nS[Cv,lc˙l+Ul∂c˙l∂T]}=-1c¯vρρT˙∂c¯v∂T+Cvc˙+UTνTDrK.

In this section, the resulting numerical method using the classical linear algebra formulation is shown. The ODE system Eq. () is assumed to be autonomous. Substituting the values of J, Eq. (), into the Rosenbrock method Eq. () yields the classic linear algebra formulation for computing a new state vector. Note that the partial derivative ft, see Sect. , will be dropped. Also, still, for box-model atmospheric chemistry systems, the equation for T will be dropped. Consequently, for atmospheric systems, no dependencies of the rate constants ki are considered in the numerical procedure, i.e., the system is assumed to be fully autonomous from the numerical point of view. Still, rate constants of photolytic reactions are updated during integration, according to current time and the specified location, i.e., zenith angle of the sun. Accordingly, this gives 29yn+1=yn+∑i=1smiu(i)ũ(i),withI-hγνTDrνeDc-1-hγνTDrK-hγc¯vρ(CvT˙+UTJcc)1+hγc¯vρ(ρ∂c¯v∂TT˙+Cvc˙+UTJcT)u(i)ũ(i)=hνTr(i)+cemis-1c¯v(i)ρ(U(i))T(νTr(i)+cemis)+∑j=1i-1diju(j)ũ(j),i=1,…,s. The values of both diagonal matrices Dr and Dc are calculated at the beginning of each time step (i.e., at y(n)), meaning that they do not form intermediates. In contrast, the reaction rates ri, the mass-averaged specific heat at constant volume c¯v,i, and the internal energy values UiT on the right-hand-side form the intermediate solution for each stage r(i)=r(i)tn+αih,yn+∑j=1i-1aiju(i)ũ(i),c¯v(i)=c¯v(i)yn+∑j=1i-1aiju(i)ũ(i),U(i)=U(i)T+∑j=1i-1aijũ(i) Note that the value for T is located in the last entry in the state vector y(n), see Eq. ().

In this work, direct linear solvers are considered to calculate the solution of the linear systems in the integration procedure. For reasons of simplicity, the linear systems are denoted as Ax=b in this section. A direct solver strategy requires the factorization of the coefficient matrix A into a lower and an upper triangular matrix L and U, where A=L⋅U. The implemented sparse solver consists of three parts, (i) the symbolic phase, where the actual factor matrix LU(A) is built and stored as separate sparse matrix as it has the same structure (i.e., non-zero pattern) in every time step, (ii) the numerical factorization within the Rosenbrock method, and (iii) a solution phase for the triangular systems. While factorizing sparse matrices, generally fill-in occurs and the factors of LU(A) become more dense than the original matrix A. However, to take advantage of the sparseness of the coefficient matrix A, the equations must be arranged in special order. To find a good ordering for the sparse matrix A, we have to determine a permutation P that minimizes the fill-in in the factors L and U. Since the problem of finding a permutation P that minimizes the amount of fill-in can be reduced to an NP-complete task , finding the optimal solution becomes computationally infeasible for larger systems due to the exponential growth of required computational resources. Yet, Cminor uses the minimum-degree ordering heuristic proposed by in a symmetric manner, meaning the pivot always lies on the diagonal to preserve existing stability. Thus, if rows i and j are swapped, then columns i and j are swapped as well. The (symbolic) elimination process is denoted in Algorithm . A is assumed to be unsymmetric and regular, which does generally not ensure that a solution exists. Nevertheless, numerical pivoting is not implemented in this approach. The numerical phase (ii) takes place once at each time step within the Rosenbrock method. By using the strategy, the growth of the number of non-zero elements is roughly bounded by the factor 2.

For multi-phase systems in a box-model, droplets of the same size will behave equally as long as the initial concentrations are equal in each droplet. If this is not the case, as is for specifying log-normally distributed aerosol, each aqueous species has to be considered once for each droplet class. A droplet class refers to a number of droplets of equal size and chemical composition. This corresponds to the concept of superdroplets commonly used in cloud microphysics .

When considering multiple droplet classes, the number of equations in atmospheric systems grows from nGas+nHenry+nAq to nGas+nHenry+nD⋅nAq, where nGas is the number of purely gaseous species, i.e., not dissolving, nHenry the number of gaseous species that may dissolve in droplets, nAq the number of aqueous species in the mechanism and nD the number of droplet classes. This is a significant increase for mechanisms with a prominent aqueous phase. For instance, a number of 50 droplet classes is common in super-droplet models. Since the equations for each droplet class are equal in structure, as the same reactions occur in every droplet class, the matrix A has a specific, repeating pattern. Examples are shown in Fig. , displaying the sparsity pattern of the iteration matrix of a sulfur oxidation mechanism in clouds for one (a) and five droplet classes (b, c). The repeating pattern, in turn, can be represented by considering the matrix for one droplet class, with vectors representing the entries of aqueous species and mass transfer terms.

By doing so, the (symbolic) matrix and its (symbolic) LU decomposition stay the same for any number of droplet classes. Therefore, only the solution phase (iii) changes from scalar operations to vector operations. These are fast and minimize indirect accesses to the respective arrays, which make up a major portion of the total solution process, and therefore also of the total simulation. Of course, since there are five droplet classes in this example, the matrix to be decomposed is (b) or (c), in Fig. , which is just a matter of ordering.

In the following, we explore whether the decomposition can be determined using only (a). As shown in (c), the vector structure can be represented by arranging all entries corresponding to aqueous species into diagonal matrix blocks. If every zero entry modification (fill-in), i.e., change of a zero entry to a non-zero entry while decomposing the matrix, is part of a new diagonal matrix block, then it is possible to work with (a) to find the decomposition of (c). Consequently, when a diagonal block matrix transforms into a fully populated matrix during the decomposition process, the vector representation becomes impractical. In such iteration matrices, the vectorized decomposition approach is no longer viable. Also, any single entry appearing off-diagonal in a diagonal matrix block prevents the vector representation of the matrix as its position would have to be stored.

Every row in the iteration matrix corresponds to an equation of a specific species, while a column resembles the influence of a specific species on the other species. This means, every row and every column relate to one specific species. By permuting the rows and columns representing Henry species, i.e., species that are gaseous but may dissolve in droplets, to the last rows and columns, the vector structure will be preserved during decomposition. Ordered like this, the iteration matrix for multi-phase mechanisms exhibits the four regions A=in-phase reactionsHenry species→aqueous or purely gaseous speciesaqueous or purely gaseous species→Henry speciesHenry species→Henry species, where purely gaseous species means not dissolving in droplets.

In the following, the preservation of the vector structure by this ordering is reasoned by an induction approach. It is shown that, having a matrix which can be represented with vector entries and which is ordered in the proposed way, a new fill-in will always occur vector-wise, preserving the possibility to represent the matrix with vector entries. The following Algorithm  generates the LU decomposition of a matrix A.

It can be seen that a necessary condition for fill-in at entry A[i,k] is 30∃j>0,j<i,j<k:A[i,j]⋅A[j,k]≠0. With this necessary condition, the logical path of the following reasoning is

fill-in → Eq. () → appropriate fill-in (vector-wise where needed).

The bottom left and upper right regions of A can in any case be represented by vector entries as the vector representation only prevents storing zeros in the upper left region. The lower right region has scalar entries anyhow, representing gaseous reactions of dissolving species (not shown in Fig. ). Therefore, it is sufficient to show that if fill-in occurs for the case of i and k both representing aqueous species, fill-in in the form of a diagonal matrix block follows.

Let i,j,k>0 be a triple of indices, for which A[i,j]⋅A[j,k]≠0 holds for a given matrix A, where the gaseous, dissolving species are permuted to the last rows and columns. If i,k≤nAq⋅nD+nGas, the entry A[i,k] is in the region of either diagonal matrix entries, i.e., aqueous species, or single-value entries, i.e., purely gaseous species. As mentioned above, this is the only important case for which preservation of the vector structure of A needs to be shown. Any entry A[i,j]≠0 comes from a reaction of species j to species i (or an equivalent previous fill-in). Since there are no reactions between purely gaseous species and aqueous species, it follows that i,j, and k all belong to the same type of species, either purely gaseous or aqueous. It also follows that fill-in can never occur between aqueous and purely gaseous species. If they are gaseous, no vector structure has to be preserved, in this case the fill-in is irrelevant. If they are aqueous, i,j, and k can be written as 31i=nGas<i+(Si-1)⋅nD+d,j=nGas<j+(Sj-1)⋅nD+d,k=nGas<k+(Sk-1)⋅nD+d, where nGas<i is the number of purely gaseous species that were permuted before species i, Si∈{1,…,nAq} is the aqueous species number disregarding droplet classes, and d∈{1,…,nD} is the droplet class in which species i is dissolved. The analagous nomenclature is used for the variables containing j and k. Note that d is the same for i,j, and k. Before fill-in, the matrix is assumed to have proper vector entries, and we re-iterate that there is no reaction from one droplet class to another. Thus, d is the same for i,j, and k. Now, w.l.o.g., it is assumed that d=1. For other values of d, analogous calculations yield the same result. The vector structure is preserved if 32∀l1,l2∈{0,…,nD-1},l1≠l2,∀j*>0,j*<i*:=i+l1,j*<k*:=k+l2:A[i*,j*]⋅A[j*,k*]=0. Under this condition, the entry A[i*,k*] will not undergo fill-in, i.e., every off-diagonal entry in the submatrix A[i:i+nD-1,k:k+nD-1] stays equal to zero. Then, the possible fill-in is a diagonal matrix, which is what needs to be shown. The updated representation of Eq. (), which is i*=i+l1=nGas<i+(Si-1)⋅nD+d+l1,k*=k+l2=nGas<k+(Sk-1)⋅nD+d+l2, helps recognizing that, since l1≠l2 by Eq. (), species i* and k* now belong to separate droplet classes, namely d+l1 and d+l2, respectively. Since the matrix is assumed to have proper vector entries before the examined fill-in, it can only have non-zero values at entries which belong to the same droplet class. Any j*>0 with j*<i* and j*<k* may correspond to an aqueous species of a specific droplet class, where either A[i*,j*] or A[j*,k*] or both are equal to zero because of the aforementioned reason, or it may correspond to a purely gaseous species, where the respective matrix entries are zero, too, because purely gaseous species do not interact with the aqueous species i* and k*. Thus, Eq. () holds.

Hence, the vector structure of A, e.g., the rightmost matrix in Fig. , is preserved during decomposition for the proposed ordering, which is to permute the gaseous dissolving species to the last rows and columns. In opposition to this, if i represents a Henry species, which is permuted between aqueous and purely gaseous ones, Eq. () is violated for l1=1, l2=0 and j*=j. In this case, the vector structure is not preserved.

For the Jacobian J, we need derivatives of Eqs. (), (), and () with respect to the rest of the variables. To minimize floating point operations, we neglect these derivatives, resulting in a sort of explicit scheme for these equations, which we assume is reasonable as they behave not as stiff as the chemical species' equations. The only exception is for the condensational equation, where we calculate the derivative with respect to the water masses, where i,j∈{1,…,nD} and nD is again the number of droplet classes. Here, ∂∂ql,jdql,idt=∂∂ql,jDi⋅(S-Seq,i)=∂Di∂ql,j⋅(S-Seq,i)+Di⋅∂(S-Seq)∂ql,j33≈Di⋅∂(S-Seq,i)∂ql,j. Since supersaturations in liquid clouds are small, we neglect (∂Di/∂ql,j)⋅(S-Seq,i), and Di summarizes all remaining factors in Eq. (). The remaining term splits up in the derivatives 34∂S∂ql,j and -∂Seq,i∂ql,j. The first term is zero since S has no direct relation to the liquid water mixing ratio. For the second term, we omit the index indicating the droplet class because for i≠j the term is zero. Also, we again summarize factors to R and K, the Raoult and the Kelvin term, respectively. Thus, 35∂∂qlSeq=∂∂ql(R⋅K)=∂∂qlql/Mwql/Mw+ns⋅exp⁡ard=R′⋅K+R⋅K′ The derivative of the Raoult term with respect to water mass is easily calculated as 36R′=nsMw(ql/Mw+ns)2. For the derivative of the Kelvin term, we note that the radius rd might indicate the radius of a droplet not only consisting of water, but also soluble and insoluble material. However, assuming ideal solution behavior for droplets of a size larger than a few nanometers, the volume of a single droplet might be calculated as V=Vw+Vrest with Vw=ql/(ρwn), n is the number of droplets (in the droplet class) per kg of air, and Vrest the volume of dissolved species. Doing so, we get 37K′=∂exp⁡ard∂rd⋅∂rd∂Vw⋅∂Vw∂ql=-ard2exp⁡ard⋅14π34π(Vw+Vrest)-23⋅1ρwn=-a4πρwnrd4exp⁡ard.

The resulting classical linear algebra scheme is 38yn+1=yn+∑i=1smiu(i)ũ(i),withI-hγνTDrνeDc-100I-hγJparcelu(i)ũ(i)=hνTr(i)+cemis+cρ˙ρq˙l(i)-gcpaw+Lvcpaq˙l(i)-q˙l(i)-ρagRaTw-pRaT2-gcpaw+Lvcpaq˙l(i)w+∑j=1i-1diju(j)ũ(j),i=1,…,s, where ũ(j) denotes the jth intermediate of the transformed parcel variables, 39q˙l(i)=diag(D(i))(S(i)⋅1-Seq(i)) and 40Jparcel=Jmm000000000-Jmm00000000000000, with 41Jmm=-diag(D)diag(R′)K+diag(R)K′. The bold letters indicate vectors containing values of all droplet classes. This represents an approximate Jacobian matrix. Note that the concept of vector entries of Sect.  is applied here (if nD>1). Wherever no index to indicate the stage value is written, the value of the first stage is taken, i.e., i=1, as the Jacobian is only calculated at the beginning of a time step. Again, the reaction rates r(i) form the intermediate solution for each stage as r(i)=rtn+αih,yn+∑j=1i-1aiju(i)ũ(i).