In the space plasma environment, radiation belts refer to torus-shaped regions surrounding Earth and other magnetized planets that are filled with highly energetic charged particles trapped in the planetary magnetic field. Since their discovery , radiation belts have been the focus of intense research due to the innumerable unknowns concerning their extremely dynamic behavior and their damaging effects on spacecraft (e.g., ). During slowly changing conditions, radiation belt particles undergo three types of periodic motion: gyration about field lines, bounce along field lines, and drift about the planet. With each periodic motion there is a corresponding adiabatic invariant, defined through the Hamiltonian action integral e.g.,chap. 10, that is only violated when the conditions are undergoing change on timescales shorter than the period. A widely adopted method to study the dynamics of radiation belts is to solve a kinetic equation describing the evolution of particle phase space density. In quasi-linear theory, this kinetic equation is usually a Fokker–Planck equation that takes the general covariant form 1∂f¯∂t=1G∂∂QαGDαβ∂f¯∂Qβ-1G∂∂QαGhαf¯+Sf¯+v, where f¯ is the phase-averaged phase space density, G=det(∂Jα∂Qβ) is the Jacobian determinant for the transformation from canonical action variables Jα (α=1,2,3) to the generalized coordinates Qα, and Dαβ, hα, S, and v are coefficients of the equation. Summation on repeated Greek indices is implied throughout this paper. In different radiation belts, the number of terms emerging on the right-hand side of Eq. () and their respective physical backgrounds may be different. For the Earth's outer radiation belt, the second and the fourth terms are usually missing; the first term represents diffusion caused by wave–particle interactions, and the third term is often a loss characterized by the particle lifetime e.g.,and the reference therein. In the low-altitude inner radiation belt where wave–particle interactions are not as significant, the first and second terms are often provided by the diffusion and dynamic friction caused by inter-particle Coulomb collisions e.g.,, and the fourth term may be a source from cosmic ray albedo neutron decay (CRAND; e.g., ). For radiation belts of the gas giants, all terms could be present (e.g., ). The first two terms may be attributable to both wave–particle interactions and inter-particle collisions and, in addition, synchrotron radiation, which is negligible in Earth's radiation belts, bleeds energy for the ultra-relativistic electrons, and thus also contributes to the second term e.g.,. The third term could represent the moon-sweeping loss, and the fourth term may come from moon volcanic activities as a plasma source e.g.,.

In some circumstances, the dependence of phase space density on certain phases φι could be discerned, and the radiation belt kinetic equation takes the form 2∂f¯∂t+φ˙ι∂f¯∂φι=1G∂∂QμGDμν∂f¯∂Qν-1G∂∂QμGhμf¯+Sf¯+v, where a dot over φι indicates its time derivative. The phase space density f¯ here is only averaged over the phases varying faster than φι, and the mutually exclusive indices ι and μ together form the complete set of α. The most common situation is perhaps the dependence of f¯ on the drift phase φ3, which in the Earth's outer radiation belt may be caused by the wave activity dependence on magnetic local time e.g., and in the inner belt by the longitudinal variation of drift shell altitude e.g.,. With the spatial derivative term on the left-hand side, Eq. () appears as a Boltzmann equation for f¯, but by Hamiltonian mechanics, the conjugating term Q˙ι(∂f¯/∂Qι) should have also appeared on the left-hand side. Its absence is due to the fact that, for particles in the radiation belt energy range, the drift-phase-dependent electric potential energy is usually negligible in the unperturbed particle Hamiltonian so that φ3 becomes a cyclic variable. Appendix  provides a more comprehensive explanation of this equation, and we will return to the general case in which φ3 is not cyclic in the “Conclusions and discussion” section.

Various numerical models have been built to solve a specific form of either Eq. () or () (e.g., , to name a few). Though their underlying numerical schemes might not restrict, these existing models have in practice been implemented with hard-coded choices of coordinates, relatively simple boundary geometry, and a roughly fixed number of equation terms; therefore, each model is applicable to a specific set of problems. This situation could become quite inconvenient when adiabatic invariants of particle motion are used as coordinates of phase space to model radiation belt dynamics, as promoted by . The reasons are twofold: first, due to their vast range of magnitude and dramatically varying resolution, adiabatic invariant coordinates often require some kind of rescaling and transformation e.g.,, specific to the problem, to be computationally efficient; second, boundary geometry becomes complicated in adiabatic invariant coordinates, which could be challenging for finite-difference methods and led to seek new coordinates from combinations of the adiabatic invariants. However, the use of adiabatic invariant coordinates is crucial for some compelling problems in the radiation belts: for example, the mechanisms of storm-time electron loss in which adiabatic modulations due to magnetic field configuration change must be separated from non-adiabatic processes and the relative significance of Earthward diffusion versus CRAND as a possible inner belt electron source with the drift shell splitting effect contributing . It is the purpose of this article to present a numerical code, named UBER (for “universal Boltzmann equation solver”), that solves Eqs. () and () in an arbitrarily user-specified coordinate system up to three dimensions, with great freedom in specifying boundary geometry and boundary conditions and with various combinations of equation terms. Therefore, it is expected that the solver can be applied to a wide spectrum of radiation belt modeling problems. More importantly, the freedom of specifying equation terms implies that, in an asymptotic manner, UBER can even solve the integro-differential kinetic equations arising from non-diffusive particle transport, such as that formulated in for nonlinear wave–particle interactions, and thereby provides a viable means to incorporate non-diffusive transport into global radiation belt modeling.

The underlying mathematical theory of the solver is stochastic differential equation (SDE) theory. The SDE method had been utilized by , , and in their modeling of radiation belts. The method is grid-free and enjoys unparalleled advantages in dealing with cross-diffusion components and complicated boundary geometry e.g.,, but it is meanwhile notorious for low efficiency ascribed to its Monte Carlo nature. In this article, we also describe specially designed numerical techniques that have enhanced the computational speed of the SDE method by an order of magnitude, thus making the solver much more affordable for large-scale simulations. Four example problems with distinct physical backgrounds are provided in this article to demonstrate the abilities and versatility of the solver.

The kinetic equations given as Eqs. () and () above are parabolic partial differential equations (PDEs). Written in the Kolmogorov backward form (see below), a parabolic PDE corresponds to a multidimensional SDE that describes the motion of an Itô stochastic process whose certain functional expectation satisfies the PDE; the PDE can then be solved by calculating path integrals of the corresponding stochastic process e.g.,.

Let us consider the following partial differential problem composed of a Kolmogorov backward equation and a general set of initial and boundary conditions: 3∂tf=12aαβ(t,x)∂α∂βf+bα(t,x)∂αf+c(t,x)f+u(t,x),4f(0,x)=g0(x),x∈Ω¯,5f(t,x)=g1(t,x),x∈∂1Ω,6γ^(x)⋅∇f-λ(t,x)f=0,x∈∂Ω\∂1Ω, where ∂t and ∂α are shorthand for the partial differentials with respect to t and the αth coordinate, respectively. In Eqs. ()–(), Ω¯ denotes the closure of the domain and ∂Ω its boundary. In particular, ∂1Ω represents the boundary pieces of the first type (Dirichlet) boundary condition, and ∂Ω\∂1Ω indicate the boundary pieces excluding those in ∂1Ω, which are of the second (Neumann, λ≡0) or the third type (Robin, λ≠0) boundary conditions. The unit vector γ^ points into Ω¯ and is not tangent to the local boundary.

The mathematical theory of SDEs establishes a relation between Eqs. ()–() and the Itô stochastic process, whose spatial positions are denoted by the random variable Xs in Ω that obeys the reflected SDE: 7dXs=b(t-s,Xs)ds+σ(t-s,Xs)⋅dWs+γ^(Xs)dks, where the dot product on the right-hand side is between a rank-2 tensor and a vector, and the parameter s runs from 0 to t so that the stochastic process retrogrades in time from t to 0. The first term on the right-hand side describes the ballistic part of its motion. The second term describes the stochastic part, with the coefficient tensor σ satisfying σ⋅σT=a (whose components are aαβ). Note that this condition does not uniquely determine σ, but all satisfying σ tensors are equivalent (Levi's theorem; ). Ws is a vector Wiener process of the same dimensions as Xs, with each dimension an independent Gaussian stochastic variable that has zero mean and variance s. The third term describes reflection of the stochastic process in the direction given by γ^ on the boundary ∂Ω\∂1Ω, and ks is a monotonic stochastic variable that only increases when the stochastic process is on that boundary to force Xs to stay in Ω¯. ks can thus be considered a measure of the time that the stochastic process spent on ∂Ω\∂1Ω and hence has the name local time. The Itô process stops either in Ω¯ when s=t or on ∂1Ω at s=τ<t.

A formal solution of the problem in Eqs. ()–() is given by the Feynman–Kac formula e.g.,: 8f(t,x)=E[Ft,x[Xs]], in which E is the expectation operator, and Ft,x[Xs] is a functional of the stochastic path Xs started from t and x that has the expression 9Ft,x[Xs]=Iτ≥tg0(Xt)exp⁡∫0tc(t-s,Xs)ds-∫0tλ(t-s,Xs)dks+Iτ<tg1(t-τ,Xτ)exp⁡∫0τc(t-s,Xs)ds-∫0τλ(t-s,Xs)dks+∫0t∧τu(t-s,Xs)exp⁡∫0sc(t-r,Xr)dr-∫0sλ(t-r,Xr)dkrds, where the symbol Iτ≥t is equal to 1 when τ≥t, which means the stochastic process has stopped in Ω¯ before it had a chance to reach ∂1Ω, and zero otherwise; t∧τ means the smaller between the two. Physically, the functional Ft,x[Xs] is a propagator of contribution carried along the stochastic path from either the initial condition or the first type boundary condition to the point of solution, and the exponential functions indicate how this contribution enhances or decays along this path.

To formally solve the Fokker–Planck equation (Eq. ) by the Feynman–Kac formula (Eq. ), it remains to transform the equation together with its proper initial and boundary conditions into the form of Eqs. ()–(). To this end, directly expanding Eq. () and collecting terms with the same differentiation order yields its Kolmogorov backward form: 10∂tf¯=Dαβ∂α∂βf¯+(∂βDαβ+Dαβ∂βln⁡G)-hα∂αf¯+S-(∂αhα+hα∂αln⁡G)f¯+v. Comparing Eq. () with Eq. () and taking Qα equivalent to xα, we thus have the correspondences of coefficients: 11a=2D,b=∇⋅D-h,c=S-∇⋅h,u=v, where in curvilinear coordinates, the divergence operator on a tensor field Γ is 12∇⋅Γ=∂αΓα…+Γα…∂αln⁡G, in which the dots stand for all other indices irrelevant to the operation, and the terms ∂αln⁡G come from summation of the Christoffel symbols in a covariant derivative e.g.,chap. 15. It is worth remarking that -h appears in the expression for b so that the Itô process travels against the advection velocity. This is indeed the case since it is time-backwards. Also, from the expression for c, divergence of the advection serves as a loss of phase space density.

Initial and boundary conditions to Eq. () are transformed as follows. For the initial condition and the first type boundary condition, values of f¯(t,x) are specified just as in Eqs. () and (). For a flux boundary condition of the form Φ=g2(t,x)f¯, we note that the outward flux Φ across a boundary is given by (n^⋅D⋅∇f¯-n^⋅hf¯), with n^ the unit inward normal vector of ∂Ω\∂1Ω. Therefore, the corresponding boundary condition is 13n^⋅D⋅∇f¯-(n^⋅h+g2)f¯=0. Comparing Eq. () with Eq. (), we determine that 14γ^=n^⋅D|n^⋅D|,λ=n^⋅h+g2|n^⋅D|. Although the SDE (Eq. ) does not prevent σ, and hence D, from being zero, the expressions in Eq. () do become singular for vanishing D on ∂Ω\∂1Ω. In the region where D vanishes, Eq. () is no longer parabolic but degenerates to an advection equation (a first-order PDE), for which imposing a Neumann or Robin boundary condition is over-determinant. In this case, we invoke on the boundary minimal diffusion in the eigen-direction of n^ so that γ^=n^, and let g2≡-n^⋅h so that λ=0, which means the advective flow is free to cross the boundary. The situation that D is finite but |n^⋅D| vanishes is considered pathological to our problem.

Up to this point, we have transformed the Fokker–Planck equation (Eq. ) and its initial and boundary conditions to the problem in Eqs. ()–() and gathered all expressions in Eqs. () and () for the constructing components of the SDE (Eq. ) as well as the functional (Eq. ). In order to solve the Boltzmann equation (Eq. ), it suffices for us to just transform the equation into the form of Eq. (). To this end, we expand the phase space by concatenating the coordinates Qμ and φι so that xα={Qμ,φι} (recall that α={μ,ι}) and introduce the new coefficients D̃αβ, h̃α, S̃, and ṽ that satisfy the following conditions: 15D̃μν=Dμν,D̃αι=0,h̃α={hμ,φ˙ι},S̃=S+∂ιφ˙ι,ṽ=v. It can be verified that Eq. () in the new xα coordinates with the new coefficients given by Eq. () transforms into Eq. () after replacing xα by Qμ and φι. The transformation (Eq. ) essentially treats φι as new dimensions of the stochastic motion, except that the stochastic part of the motion in these dimensions is identically zero. A new type of boundary condition might emerge for problems involving Eq. (), which is the periodic boundary condition for the phases φι. From the viewpoint of stochastic motion, though, such periodicity is not really a boundary but rather a topology of Ω. The treatment of periodic boundary condition will be exemplified in the third problem in Sect.  below.

To summarize this section, the above mathematical theory allows us to fully define a PDE problem involving Eq. () or () in an arbitrary coordinate system given the input functions and equations as listed in Table , which can be either analytical or numerical in the UBER code. The equation terms may be freely turned off by setting their corresponding coefficients to zero. The number of boundary pieces is totally up to choice, which can even be zero to put the boundary at infinity. The boundary geometry may be time-variable for boundary pieces in ∂1Ω in the UBER code, but it must be fixed for those in ∂Ω\∂1Ω. Solutions of this problem are obtained once we find a way to evaluate the functional in Eq. () for a realization of a stochastic path and to estimate the expectation of the functional. These numerical techniques are the subject of the next section.

We give an outline of the algorithms used by the UBER code in this section, with an emphasis on the techniques that improve both its accuracy and efficiency. Lower-level numerical techniques, such as the generation of pseudo-random variables, linear algebraic operations, and parallelized computation, are based on the works presented in . The general idea for numerically implementing the SDE method is as follows: (i) for a given spatiotemporal position (t,x) for which an equation solution is wanted, a number of stochastic paths starting from this common position are simulated; (ii) for each stochastic path, its functional value is evaluated by the path integrals as in Eq. (); and (iii) from these sampled functional values, their expectation is estimated, and this gives the solution at (t,x). Therefore, the SDE method is essentially a Monte Carlo method. It does not rely on a computational grid and is able to solve the problem locally. However, on many occasions it is still worth obtaining global solutions on a grid so that the solutions at time stamp Ti may be used as the initial condition for the solutions at Ti+1, analogous to the idea of layer methods e.g.,. In this way, the stochastic processes need only to be simulated for a short duration of t=Ti+1-Ti to obtain the new solutions, for which the calculation of functional expectation would converge much faster than those simulated for the full length t=Ti+1. The only operation on this grid would be interpolation and possibly extrapolation; therefore, unlike in layer methods, the grid can be irregular and thus allows for the use of sophisticated interpolation libraries on irregular grids, although in the present version only a nonuniform Cartesian grid (in the given coordinate system) is implemented with user-specified nodes.

Integration of the SDE (Eq. ) employs the Euler–Maruyama scheme that is order 1 for weak convergence problems such as ours, meaning that when only the statistical distribution of stochastic paths matters but not the individual path, the expectation of the schematic error is proportional to the first power of the time step size . To further reduce the schematic error, an adaptive time step size is used in UBER. It can be shown that e.g., the root mean square (rms) distance an Itô stochastic process travels in infinitesimal time ds is 16dX¯s=tr(a)ds. Numerically, the first-order contribution from |bΔs| cannot be neglected due to the finite Δs. Therefore, we prescribe a desired rms spatial step size ΔX¯s, which is sufficiently small compared to the size of Ω and any scale length of the equation coefficients, and then choose the smaller Δs inferred from either Eq. () or ΔX¯s=|bΔs| at every step of integration as the adaptive step size. This scheme evidently reduces to a simple adaptive Euler scheme for integrating ordinary differential equations when a approaches zero.

Oblique reflection of the stochastic process on ∂Ω\∂1Ω and the calculation of dks follow the projected half-space algorithm presented in , which is also order 1 in the weak convergence sense. The idea is that, for an exact half-space boundary, ks can be proven to share the same probabilistic distribution with a composite stochastic variable involving Ws, coefficients of the SDE, the normal vector n^, and an independent exponential random variable with parameter (2s)-1 ; therefore, ks can be explicitly calculated by these known quantities. For general smooth boundary geometry, an additional contribution to dks may also come from possible projection along the γ^ direction needed to keep the stochastic process within domain. With dks obtained and the SDE (Eq. ) integrated, the functional (Eq. ) can be readily evaluated by an ordinary numerical integration technique implemented along the realized stochastic path.

Expectation of the functionals can be estimated, in principle, from an arithmetic mean of a number N of sampled stochastic path integrals. The error of this estimation, ϵ=|E[Ft,x]-〈F̃t,x〉|, where a tilde is used to indicate a numerical realization in this section and 〈⋯〉 indicates averaging over samples, can be estimated by dividing the simulation of stochastic processes into batches . Although the probabilistic distribution of individual F̃t,x is generally far from normal and largely unknown, that of the batch-wise mean of F̃t,x approaches a Gaussian for a large enough sample number per batch due to the central limit theorem, and a confidence interval can thereby be calculated for the batch-wise means using the Student's t distribution e.g.,. We use this confidence interval as an approximation to ϵ. In this way, UBER adaptively stops simulating more batches of stochastic processes when the estimated error meets a prescribed tolerance.

In typical radiation belt problems, the functional values from various stochastic paths may differ by orders of magnitude, and hence their contributions to the arithmetic mean also differ by orders of magnitude, whereas their computational efforts are of the same order. Therefore, straightforward calculation of their arithmetic mean could result in extremely slow convergence with N and squander computational power. To reduce statistical variance in this procedure, a process-splitting technique is developed based on the idea of importance sampling, i.e., to make “denser” sampling in more important “regions”. In conventional Monte Carlo methods, the “region” is an “area” in a parameter space, and importance sampling effectively splits one sample point therein that would have made a huge contribution to the calculation into many sample points nearby, while weights of these samples are reduced accordingly to keep the probabilistic distribution of samples unbiased e.g.,. But unlike conventional Monte Carlo methods, the samples in the SDE method are paths which belong to a functional space. To still implement this idea, we split the stochastic path when it is projected to contribute a large functional value.

Figure  gives an illustration of this technique in a t⊗R2 space. As a stochastic path being integrated from point P, the functional value of the entire path (from s=0 to s=t) is continuously predicted based on the partial path that has been realized. This projected functional value is compared to the value of some quantile (e.g., the 80th percentile) statistically derived from all previously completed stochastic paths starting from the same position. When at some place Q the projected functional value falls above this quantile, the stochastic process is deemed to make a significant contribution to the arithmetic mean. It is then split into a number of child processes at Q, and each child process traces down an independent path thereafter. These child paths, together with their common parent path segment PQ, hence constitute “nearby samples” in the functional space. This procedure can be further iterated if the projected functional value later falls into an even higher quantile (e.g., the 90th percentile), as shown at R. After all procedures finish, the eventual result is a tree structure of stochastic paths rooted at P. For the illustration in Fig. , the actual functional value of the path PQA will be weighted by 1/2, and those of PQRB and PQRC will be weighted by 1/4, when calculating their contributions to the mean. In the UBER code, a practical choice for the number of children at each splitting is 4, and that for the upper limit of offspring generations is 3, so that a stochastic process can be split into a maximum of 43=64 processes. Effects of the process-splitting technique are studied in the first problem in the next section.

It still remains to find a method to project the functional value of a stochastic path when it is only partially realized. For this purpose, we insert a break point at s=s′∈(0,t) to the integrations in Eq. () and see how it transforms. We simplify the situation by only considering the stochastic processes stopping in Ω¯ for the moment and denote the following functional integrals: 17U0t=exp⁡∫0tcds-∫0tλdks,18V0t=∫0tuexp⁡∫0scdr-∫0sλdkrds, in which the integrand functions c, λ, and u are as those in Eq. (). Then, the functional Ft,x with the above presumptions and notations is transformed as 19Ft,x=g0(Xt)U0t+V0t=g0(Xt)U0s′Us′t+V0s′+U0s′Vs′t=g0(Xt)Us′t+Vs′tU0s′+V0s′=Ft-s′,Xs′U0s′+V0s′, where Ft-s′,Xs′ is the functional for a stochastic process that starts from the break point (t-s′,Xs′) and continues until s = t.

Suppose that a partial path has been realized up to s=s′. From it we can readily evaluate U0s′ and V0s′ in Eq. () and therefore need an estimated F¯t-s′,Xs′ to project the functional value F¯t,x, where a bar is put over all unrealized entities. Specifically, we would need these three estimates: X¯t, U¯s′t, and V¯s′t. In principle, a good estimation of X¯t is given by integrating along the streamline of the b(t-s,x) field through Xs′ until s=t, which means projecting for X¯t along the ballistic trajectory of motion while ignoring all the stochasticity since the Wiener process has zero mean. However, this integration is not much cheaper than the realization of Ft-s′,Xs′ itself and is thus unaffordable. In anticipation that the total time length t would not be too large, especially when using a solution grid, and that b(t-s,x) would not vary drastically in this time interval, mapping X¯t along the constant vector b(t-s′,Xs′) is a good enough but much cheaper approximation. If X¯t is mapped out of Ω¯ so that g0(X¯t) is unable to be evaluated, the particular stochastic process is then disabled from splitting.

The functional values U¯s′t and V¯s′t are estimated by assuming that, for all possible stochastic paths belonging to the same solution point, there are mean functions c¯, λ¯, and u¯ that are independent of time and that the mean local time is proportional to the total time length of the stochastic process so that k¯s=k¯s with k¯ the proportionality constant. Under these assumptions, U¯s′t and V¯s′t can be expressed by 20U¯s′t=exp⁡c¯∫s′tds-λ¯∫s′tdks=exp⁡(c¯t-λ¯k¯t)-(c¯s′-λ¯k¯s′)=U¯0texp⁡s′tln⁡U¯0t, and 21V¯s′t=V¯0tU¯0t-1U¯0t-exp⁡s′tln⁡U¯0t, if U¯0t≠1, or by 22U¯s′t=1,23V¯s′t=V¯0t1-s′t, if U¯0t=1. The values of U¯0t and V¯0t can be well estimated, respectively, by the medians of Ũ0t and Ṽ0t that are obtained from all previously completed stochastic paths. Medians are preferred to means here because the probabilistic distributions of these functionals are usually very skewed and heavy-tailed. This projection mechanism would become statistically more accurate with more stochastic processes having been simulated.

Four example problems are provided in this section. In the first problem, we solve a Fokker–Planck equation with two source terms, one proportional to the unknown function and the other independent of the unknown function, in both a spherical coordinate system and a Cartesian coordinate system. Effects of the process-splitting technique are analyzed in this example. In the second problem, an advection-dominated Fokker–Planck equation is considered. We further show that, even for a pure advection equation, the UBER code still gives the correct solutions, although it is not designed for such an equation and may not be the most efficient method. Code behavior with advection equations is further studied in the third problem in which the treatment of periodic boundary condition is also illustrated. In the last problem, we simulate the Earth's inner radiation belt by solving its Boltzmann equation involving realistic pitch-angle diffusion and CRAND source.

In conclusion, we have built a numerical solver for the general form of kinetic equations that appear in radiation belt studies. Based on the SDE method, the solver is coded to work in arbitrarily provided coordinate systems of up to three dimensions with user-specified boundary geometry, boundary conditions, and equation terms. We have also designed adaptive algorithms and a variance reduction technique for the SDE method, which enhanced its computational speed by 1 order of magnitude in our test. The example problems in this article demonstrated the solver's versatility and robustness in dealing with a range of problems that might each require a different solver in other methods. The solver, named UBER, has been programmed into a Fortran library that can be easily incorporated with other more complicated space physics models.

The strengths of the SDE method lie in its abilities to solve problems that are difficult for other methods, but not in its speed. Even with the presented improvements, the SDE method is in general nowhere near the finite-difference counterparts in terms of speed due to its Monte Carlo nature. However, Monte Carlo methods are perfectly parallelized for no communication among parallel tasks; therefore, the UBER code gains handsome speedup from parallel computation. Figure  displays the UBER code speedup, measured as the ratio between wall clock time of serial and parallel executions, against different numbers of parallel threads employed in solving Problem 1 in spherical coordinates. For a fixed amount of computation, speedup is limited by the portion of work that cannot be parallelized; this is described by Amdahl's law inset formula in Fig. ;, which fits the numerical experiment data with parameter p=0.85, meaning that effectively 85 % of the total work has been parallelized. With 86 threads, the code approaches its theoretical maximum speedup of 6.7 for this problem, which uses about 620 s of wall clock time on a Linux server. The referenced finite-difference code uses 65 s in serial execution to solve the problem on the same server. In other words, the UBER code, in its ideal parallelization, is an order of magnitude slower than serial finite-difference methods, and it is also roughly the case for Problem 4 (Zheng Xiang, personal communication, 2021) for which the UBER code spends 22 min to simulate one drift period. This conclusion, however, is rather qualitative because in these comparisons different methods have used quite different grids. Finite-difference methods generally require a much finer grid than the SDE method in order to deal with short-wavelength components of the solution, and time spacing is restricted accordingly by the Courant–Friedrichs–Lewy condition e.g.,. On the other hand, the SDE method affords a grid that is only concentrated in interesting areas as long as global interpolation can be warranted, and the time stamps for dumping solutions are up to user choice. Moreover, if only local solutions are sought over a short time period, the SDE method could totally disregard the uninteresting solutions and might be more efficient. For radiation belt simulations in particular, finite-difference models usually rely on the operator splitting technique e.g., to solve tensorial diffusion in three dimensions, which involves frequent interpolation between two sets of grids and hence greatly hinders speed, whereas problem dimensionality has little effect on the SDE method.

Several other forms of radiation belt kinetic equations should also be solvable by the method presented in this article. In formulating the Boltzmann equation (Eq. ), we have assumed that the unperturbed particle Hamiltonian H0 is independent of phases of particle motion. For lower-energy ring current particles, the convective electric field potential energy is not negligible in their Hamiltonian, and therefore H0 would be dependent on the drift phase. As such, expanding the Poisson bracket [f¯,H0] on the left-hand side of the Boltzmann equation will result in additional terms involving partial differentials with respect to the generalized momenta Qι (see Appendix ). For a radiation belt model including ring current particles, the general form of the Boltzmann equation will be 42∂f¯∂t+φ˙ι∂f¯∂φι+Q˙ι∂f¯∂Qι=…, in which the omitted right-hand side is exactly the same as that of Eq. (). The Boltzmann equations of the so-called four-dimensional radiation belt models, such as the CIMI model , the VERB-4D model , and the K2 MHD-particle model , are of this type. Similar to the treatment of Eq. (), Eq. () can be obtained from Eq. () by introducing the new coordinates xξ={Qμ,Qι,φι}≡{xμ,xκ}, which enlarges the index set from α to ξ, and performing the following transformation of equation coefficients: 43D̃μν=Dμν,D̃ξκ=0,h̃ξ={hμ,x˙κ},S̃=S+∂κx˙κ+Q˙ι∂ιln⁡G,ṽ=v. Therefore, the Boltzmann equation (Eq. ) can also be solved by the method presented in this article in principle. However, such four-dimensional simulations are beyond the current scope of the UBER code since it is only coded for up to three dimensions in space.

Nonlinear evolution of phase space density occurs when the particle scatterings are not only small-scale but also large-scale, usually as a result of trapping by intense plasma waves e.g.,. In this case, the right-hand side of the kinetic equation must include terms of nonlocal transport of phase space density by these large-scale scatterings, and the equation is formulated as 44∂f¯∂t=1G∂∂QαGDαβ∂f¯∂Qβ-1G∂∂QαGhαf¯-∫PQ→Q̃G̃dQ̃αf¯+∫PQ̃→Qf̃G̃dQ̃α, in which f̃ is a shorthand for the function f¯(t,Q̃α), and G̃ is the Jacobian determinant evaluated at Q̃α. With nonlinear wave–particle interactions, the phase bunching effect gives rise to advection characterized by the coefficients hα. The function PQ→Q̃ is the trapping probability density per unit time from Qα to Q̃α; that is, particles are trapped by the wave field at Qα and subsequently escape from trapping at Q̃α. This is considered a known function which can be evaluated from single particle behaviors by either perturbation theory of Hamiltonian mechanics e.g., or test-particle simulations e.g.,. Note that, since the unknown function is contained in the last integral term, Eq. () is an integro-differential equation. However, formal similarity between Eq. () and the Fokker–Planck equation (Eq. ) suggests that an asymptotic solution of Eq. () may be achieved by Taylor expanding f̃ as 45f̃=f̃0+f̃0′t+…, where f̃0=f¯(0,Q̃α), and f̃0′ indicates its time-derivative function evaluated at t=0. When applying the SDE method with a solution grid, the functions f̃0 and f̃0′ can be obtained from solutions of previous time stamps. Then, by defining the following coefficients 46S(Qα)=-∫PQ→Q̃G̃dQ̃α,47v(t,Qα)=∫PQ̃→Qf̃0G̃dQ̃α+t∫PQ̃→Qf̃0′G̃dQ̃α+…, which are now known functions, Eq. () is transformed into the form of Eq. () and is readily solvable by the UBER code. In this way, simulations of nonlinear wave–particle interactions in radiation belts could hence be unified with well-developed simulations in quasi-linear theory.