CONVERGENCE OF A NUMERICAL METHOD FOR SOLVING DISCONTINUOUS FOKKER PLANCK EQUATIONS

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1 SIAM J NUMER ANAL Vol 45, No 4, pp c 007 Society for Industrial and Applied Mathematics CONVERGENCE OF A NUMERICAL METHOD FOR SOLVING DISCONTINUOUS FOKKER PLANCK EQUATIONS HONGYUN WANG Abstract In studies of molecular motors, the stochastic motion is modeled using the Langevin equation If we consider an ensemble of motors, the probability density is governed by the corresponding Fokker Planck equation Average quantities, such as average velocity, effective diffusion coefficient, and randomness parameter, can be calculated from the probability density A numerical method was previously developed to solve Fokker Planck equations [H Wang, C Peskin, and T Elston, J Theoret Biol, 003, pp 49 5] It preserves detailed balance, which ensures that if the system is forced to an equilibrium, the numerical solution will be the same as the Boltzmann distribution Here we study the convergence of this numerical method when the potential has a finite number of discontinuities at half numerical grid points We prove that this numerical method is stable and is consistent with the differential equation Based on the consistency analysis, we propose a modified version of this numerical method to eliminate the first order error term caused by the discontinuity We also show that in the presence of discontinuities, detailed balance is a necessary condition for converging to the correct solution This explains why the central difference method converges to a wrong solution Key words Fokker Planck equation, detailed balance, consistency, stability, convergence AMS subect classification 65M DOI 037/ Introduction Molecular motors operate in an environment dominated by thermal fluctuations [] In general, a molecular motor has many internal and external degrees of freedom Of these degrees of freedom, there is one associated with the main function of the motor, its unidirectional motion For example, the γ shaft of an F ATPase rotates with respect to the hexamer formed by three pairs of α and β subunits [, 3, 4], and a kinesin dimer walks along a microtubule [5, 6] In studies of molecular motors, it is natural to follow the motor along the dimension of its unidirectional motion [7, 8, 9] The effects of the other degrees of freedom are modeled in the mean field potential affecting the unidirectional motion To introduce the governing equations for molecular motors, we start with the onedimensional motion of a small particle in a fluid environment subect to a potential, V x, where x is the coordinate along the dimension of motion The particle is subect to the viscous drag force, the force derived from the potential, and the Brownian force Both the drag force and the Brownian force are the results of bombardments by surrounding fluid molecules The drag force is the mean, and the Brownian force is the fluctuation part of the random force caused by bombardments The particle is governed by Newton s second law: m dv dt = ζv V x+ dw t k B Tζ, dt where m is the mass and v the velocity of the particle In, W t is the Weiner process The drag force on the particle, ζv, is proportional to the velocity, and ζ is Received by the editors September, 005; accepted for publication in revised form January 6, 007; published electronically July 8, 007 This work was partially supported by National Science Foundation grant DMS Department of Applied Mathematics and Statistics, Mail Stop SOE, University of California Santa Cruz, Santa Cruz, CA hongwang@amsucscedu 45

2 46 HONGYUN WANG called the drag coefficient The magnitude of the Brownian force is related to the drag coefficient and is given by k B Tζ, which is a result of the fluctuation-dissipation theorem [, 3] Here k B is the Boltzmann constant and T the absolute temperature For a bead of radius a, the drag coefficient and the mass, respectively, are [] ζ =6πηa, m = 4 3 πρa3, where ρ is the density and η the viscosity of the surrounding fluid Equation can be written as dv 3 dt = ζ [ v m ζ V x+ ] dw t D, dt where D = k BT ζ m 4 ζ is the diffusion constant [] It is important to notice that the ratio is inversely proportional to the square of the radius of particle Thus, for a small particle, 5 v = ζ m = 9η ρa = O a ζ m is very large In this case, 3 is well approximated by [ ζ V x+ D ] dw t dt The reduction from 3 to 5 in the limit of large ζ m is called the Kramers Smoluchowski approximation Writing 5 as a stochastic differential equation for x, wehave 6 dx dt = ζ V x+ dw t D dt This is the Langevin equation without the inertia term, governing the stochastic motion of a small particle subect to potential V x [] In molecular motors, the mechanical motion is coupled to the chemical reaction The general mathematical framework used in modeling molecular motors is a system of coupled Langevin equations Each Langevin equation in the coupled system has the form of 6 with a periodic potential V S x, where S represents the current chemical state of the motor system [7, 9, 4]: 7 dx dt = ζ V Sx+ dw t D dt Here S N, and N is the number of possible chemical states of the motor system The period of these potentials is usually determined by the step size of the motor For example, a kinesin dimer walks on a microtubule with 8-nm steps [6] The chemical reaction of the motor system the stochastic umping of the motor system among the chemical states is governed by a discrete Markov process a ump process The motor behavior such as the average velocity can be calculated by following the stochastic evolution mechanical motion and chemical reaction of the motor in Monte Carlo simulations However, results obtained with Monte Carlo simulations have statistical errors and converge very slowly If we calculate the ensemble average

3 CONVERGENCE OF A NUMERICAL METHOD 47 by following a large number of motors, then the statistical error is inversely proportional to the square root of the number of motors in the ensemble Furthermore, when the potential ψx is not smooth, there are numerical difficulties in Monte Carlo simulations Fortunately, average quantities can be calculated more efficiently by following the probability density of the motor Let us consider an ensemble of motors, each evolving in time independently and stochastically according to Langevin equation 7 Let ρ S x, t be the probability density that the motor is at position x and in chemical state S at time t The time evolution of ρ S x, t is governed by the Fokker Planck equation corresponding to Langevin equation 7 []: 8 ρ S t = D [ V S x x k B T ρ S + ρ ] S + x N k S xρ, = S =,,,N, where, for S, k S x is the chemical transition rate from state to state S k S S x is the total rate of umping out of state S and is given by 9 k S S x S k S x A simple way to model molecular motors is to average V S x over all chemical states weighted by the steady state probability density functions of these states [0] Let ψ x be the weighted average of V S x over all chemical states: 0 ψ x N S= ρ Sx N ρ S xv Sx ψx can be viewed as the motor s mean field free energy landscape The mechanical motion of the motor can be modeled using Langevin equation 6 with potential ψx Let L denote the period of V S x We immediately see that ψ x is also periodic with period L However, ψx may not be periodic As a matter of fact, for a molecular motor undergoing a unidirectional motion powered by a chemical reaction, ψx must not be periodic If ψx is periodic, then there is no energy available to drive the motor forward because there is no free energy change going from one period to the next For the motor to go forward, there must be a free energy drop going from one period to the next Since ψ x is periodic, the energy landscape ψx isa tilted periodic potential: S= ψx + L =ψx Δψ, where Δψ >0 is the energy made available from the chemical reaction to drive the motor forward in one period An example of tilted periodic potential is shown in Figure This is also the potential we will use in numerical simulations in section 7 If the energy landscape ψx is simply a constant slope downhill, then the energy Δψ is utilized uniformly in one period to generate a constant motor force If the slope of ψx is not a constant, then the motor force varies with the motor position within one period As shown in Figure, the energy landscape ψx may not be monotonic In that case, the motor depends on the Brownian fluctuations from the surrounding fluid to get over the free energy barrier Of course, the energy source for driving the motor forward eventually comes from the free energy drop Δψ, which rectifies

4 48 HONGYUN WANG 6 4 ψx 0 - Δψ -4 x - 0 Fig Graph of the tilted periodic potential used in numerical simulations in section 7 the forward fluctuations This mechanism of driving the motor forward by rectifying thermal fluctuations is called ratchet [] If we use the mean field potential 0, then the mechanical motion of the motor is governed by the Langevin equation: dx dt = ζ ψ x+ dw t D dt Equation has been used in studies of motors [4, 5] In addition to its mathematical simplicity, another advantage of using is that the energy landscape ψx can be extracted from single molecule experimental data [0] The Fokker Planck equation corresponding to Langevin equation is [] ρ 3 t = D [ ψ x x k B T ρ + ρ ] x In [6], a robust numerical method hereafter referred to as Method was designed for solving Fokker Planck equations 8 and 3 When the potential is smooth, the proof of convergence of Method is straightforward [6] But that does not provide an accurate theoretical explanation for the robust performance of Method The strength of Method is that it works fairly well even if the potential is discontinuous In this paper, we are going to prove the convergence of Method for the model equation 3 when the potential is piecewise smooth and has a finite number of discontinuities at half numerical grid points First, we nondimensionalize 3 The dimensionless independent variables and functions are defined as x = x L, t = t D L, ψ x =ψx k B T, ρ x =ρxl, Δ ψ =Δψ k B T Since we are going to work with the dimensionless variables and functions, let us drop from the notations The dimensionless version of 3 is ρ 4 t = [ ψ xρ + ρ ], x x

5 CONVERGENCE OF A NUMERICAL METHOD 49 where ψx satisfies ψx +=ψx Δψ, Δψ >0 Equation 4 can be viewed as a special case of 3 with L =,D =, and k B T = The rest of the paper is organized as follows In section, we discuss the two conditions that the exact solution of a discontinuous Fokker Planck equation must satisfy at a discontinuity We derive the two conditions for the exact solution at a discontinuity by rewriting the Fokker Planck equation as a heat equation with discontinuous heat conductivity and discontinuous specific heat capacity The first condition is the continuity of the heat flux, which corresponds to the conservation of heat The second condition is the continuity of the temperature, which follows from the regularizing properties of the heat equation In section 3, we describe the construction and properties of Method developed in [6] The most important property of Method is that it preserves detailed balance In section 4, we prove that Method is stable with respect to a norm that is equivalent to the -norm We start by showing that the steady state solution of the half discrete method is unique and all positive This allows us to define a weighted -norm using the steady state solution as the weight function We proceed to show that the steady state solution is bounded away from 0 and from infinity, independent of the numerical grid size It follows that the weighted -norm is equivalent to the standard -norm We then show that with respect to the weighted -norm, Method is unconditionally stable In section 5, we analyze the consistency of Method when the potential is discontinuous We show that away from the discontinuity, the local truncation error of Method on the exact solution is Okk +h At the discontinuity, the local truncation error on the exact solution is O However, if we perturb the exact solution by a term of the order Oh, then the local truncation error on the perturbed solution is Okk + h Once we have both the stability and consistency, the Lax equivalence theorem [7] implies that Method converges to the correct solution of the differential equation In section 6, we propose a modified version of Method to eliminate the first order error term caused by the discontinuity hereafter referred to as Method As a result, the modified method Method is second order accurate even in the presence of discontinuities In section 7, we carry out numerical simulations using a discontinuous potential to compare the performance of the central difference method, Method, and Method The central difference method converges to a wrong solution Both Method and Method converge to the correct solution We also show that in the presence of discontinuities, detailed balance is a necessary condition for converging to the correct solution This explains the defect of the central difference method Exact solution of a discontinuous Fokker Planck equation and conditions at the discontinuity In this section, we discuss the two conditions that the exact solution of a discontinuous Fokker Planck equation must satisfy at a discontinuity We derive the two conditions for the exact solution at a discontinuity by rewriting the Fokker Planck equation as a heat equation with discontinuous heat conductivity and discontinuous specific heat capacity The first condition is the continuity of the heat flux The second condition is the continuity of the temperature We study the case where potential ψx is piecewise smooth and has a finite number of discontinuities in one period Without loss of generality, we assume that there is only one discontinuity at x d in [0, ] More specifically, we assume that ψx is two smooth functions connected by the discontinuity That is, ψx is smooth in

6 430 HONGYUN WANG [0,x d ]ifψx d is redefined as ψx d = lim x x ψx, and ψx is smooth in [x d, ] if d ψx d is redefined as ψx d = lim x x + ψx For simplicity, in this paper a smooth d function means it is infinitely differentiable, and so we can use as many terms of its Taylor expansion as we want When ψx is discontinuous at x d, ψ x is not a regular function If the system is brought to an equilibrium, the equilibrium solution is given by the Boltzmann distribution: ρx = Z e ψx, Z = 0 e ψx dx, which is discontinuous at x d Thus, we should expect ρx, t to be discontinuous as a function of x at x d In modeling molecular motors, a discontinuous potential is simply a mathematical abstraction In reality, the discontinuity represents a very narrow transition region in which the potential is smooth but changes dramatically When the potential is smooth, we can rewrite Fokker Planck equation 4 as ρ t = e ψx eψx ρ x x Let us introduce ux, t e ψx ρx, t The equation above can be written in the form e ψx u t = e x ψx u Equation has the form of a heat equation In, ux, t can be viewed as the temperature, e ψx on the right-hand side as the heat conductivity, e ψx on the left-hand side as the specific heat capacity, and e ψx ux, t =ρx, t as the heat Equation is equivalent to Fokker Planck equation 4 when the potential is smooth So it is natural for us to use the exact solution of to define the exact solution of Fokker Planck equation 4 when the potential is discontinuous The biggest advantage of using is that we can avoid the nonconservative product ψ xρx, t in Fokker Planck equation 4 When both ψx and ρx, t are discontinuous, it is highly nontrivial to interpret the nonconservative product ψ xρx, t in Fokker Planck equation 4 for example, in [3], nonconservative product of the form dw dx gw is defined as a Borel measure In, when ψx is discontinuous, both the heat conductivity and the specific heat capacity are discontinuous Away from the discontinuity, the exact solution of satisfies differential equation in the classical sense At the discontinuity, the exact solution of is constrained by two conditions The first condition is the continuity of the heat flux, which corresponds to the conservation of heat The second condition is the continuity of the temperature, which follows from the regularizing properties of the heat equation The continuity of the temperature also reflects the physical nature of the heat conduction process: temperature gradient is relaxed by the heat flow that is induced by the temperature gradient In particular, any isolated discontinuity in temperature will be removed immediately by heat conduction Now we write the two conditions in terms of ρx, t and ψx The first condition continuity of heat flux is 3 ψ xρ + ρ x x=x d = x ψ xρ + ρ x x=x + d

7 CONVERGENCE OF A NUMERICAL METHOD 43 For Fokker Planck equation 4, condition 3 means that the probability flux into the discontinuity is the same as the probability flux out of the discontinuity ie, the conservation of probability at the discontinuity, which corresponds to the wellknown Rankine Hugoniot condition for weak solutions of hyperbolic equations [4] The second condition continuity of temperature is 4 e ψx ρx, t x=x d = e ψx ρx, t x=x + d Equations 3 and 4 are the two conditions that the exact solution of a discontinuous Fokker Planck equation must satisfy at the discontinuity If a numerical method is based on conservation of probability, then the numerical solution will automatically satisfy condition 3 As we will see in section 7, condition 4 is related to detailed balance If a numerical method does not preserve detailed balance, then the numerical solution may converge to a wrong solution that does not satisfy condition 4 3 The numerical method In this section, we summarize Method proposed in [6] In the spatial discretization of 4, we divide the period [0, ] into M subintervals of size h = /M Each subinterval is represented by its center a site, and the numerical grid is formed as h = M, x = h + h, x / = h + h Since the underlying stochastic evolution is a continuous Markov process, we discretize it as a ump process discrete Markov process The idea of using a ump process on discrete sites to approximate a continuous Markov process was originated in [8] and in an unpublished result by C Peskin As shown in Figure 3, in the spatial discretization, subinterval is [ x /,x +/ ] and its center is x The motor system can reside only on a set of discrete sites {x } In a single ump, it can ump only to one of the two adacent sites Let h p t be the probability that the motor system is at site x at time t in the ump process p t can be viewed as 3 p t h x+/ x / ρx, tdx ρx,t Let F +/ be the rate of umping from x to x + forward ump and B +/ the rate of umping from x + to x backward ump The numerical probability flux through x +/ is 3 J +/ = h F +/ p B +/ p + The time evolution of p t is governed by the conservation of probability: 33 dp dt = J / J +/ h =F / p B / p F +/ p B +/ p + Before we describe how the ump rates F +/ and B +/ are calculated in Method, we would like to point out that 33 is a very general framework It can even accommodate the central difference method, which can be cast into the form of 33 with

8 43 HONGYUN WANG F -/ F +/ x - x x + x x -3/ x -/ x +/ x +3/ B -/ B +/ Fig 3 Spatial discretization of 4 The motor system is restricted to a set of discrete sites {x } and can ump only to adacent sites ump rates 34 where F CD +/ = h δψ +/ B CD +/ = h + δψ +/,, 35 δψ +/ = ψx + ψx Notice, however, that the ump rates 34 associated with the central difference method may be negative Even in the limit of h 0, the ump rates 34 may be negative when the potential ψx is discontinuous This explains why the central difference method converges to a wrong solution as we will see in section 7 In Method [6], the ump rates are calculated based on local approximate solutions In calculating F +/ and B +/, we make two assumptions: In [x /,x +3/ ], potential ψx is linear with slope δψ +/ /h This assumption is to make the method simple and easy to implement Under this assumption, the potential in [x /,x +3/ ] is given by 36 ψx =C + δψ +/ x, h where δψ +/ is defined in 35 Let ρ +/ x be the steady state solution of 4 in [x /,x +3/ ] with linear potential 36 and subect to the condition 37 h h x+/ x / ρ +/ xdx = p, x+3/ x +/ ρ +/ xdx = p + In the ump process, the probability flux through x +/ is given by that of ρ +/ x This assumption is a key component of Method [6] Instead of using the Taylor expansion, the numerical approximation is based on local approximate solutions The consequence of this approach is that detailed balance is preserved, and Method works well even if the potential is discontinuous

9 CONVERGENCE OF A NUMERICAL METHOD 433 The probability flux of ρ +/ x is derived in [6] and is given by 38 J = h δψ +/ p e δψ e δψ +/ p + +/ Comparing the theoretical flux 38 with the numerical flux 3, we immediately obtain F +/ = δψ +/ h e δψ, +/ B +/ = δψ +/ e δψ +/ 39 h e δψ, +/ where δψ +/ = ψx + ψx, as defined in 35 It is important to notice that the ump rates 39 are always positive It is straightforward to verify that F +/ and B +/ are both positive when δψ +/ 0 When ψ +/ =0,wehave F +/ = lim δψ 0 B +/ = lim δψ 0 h δψ e δψ = h, δψ e δψ h e δψ = h The property that the ump rates given in 39 are always positive will be used in the stability analysis below The ump rates given in 39 also satisfy detailed balance [6]: 30 F +/ B +/ = e ψx ψx+ In the time dimension, 33 is discretized using a Crank Nicolson-type method [9] Let p n be the numerical approximation for p nk, where k is the time step The fully discrete method is 3 p n+ { = p n p n + k F + pn+ / p n F + pn+ +/ p n B + pn+ / B +/ p n + + pn+ + } In the calculation of average velocity and/or effective diffusion coefficient, 4 is solved with the periodic boundary condition [6] In the analysis of subsequent sections, we always assume the periodic boundary condition: p n = p n +M, ψx +=ψx Δψ We will prove the stability and consistency of 3 when potential ψx is piecewise smooth and has a finite number of discontinuities at half numerical grid points 4 Stability of the numerical method In this section, we prove the stability of Method [6] We first show that the steady state solution of the half discrete method 33 is unique and is all positive Furthermore, we show that the maximum of the steady state solution is bounded by the minimum multiplied by a constant that

10 434 HONGYUN WANG is determined by the underlying physical problem but is independent of the numerical grid size Thus, we can define a weighted -norm using the steady state solution as the weighting function, and the weighted -norm so defined is equivalent to the -norm Then we prove that the fully discrete method 3 is unconditionally stable with respect to the weighted -norm For the convenience of mathematical discussion, we introduce vector notations for numerical solutions in one period: p n p n,p n,, p n M, q q,q,, q M, r r,r,, r M Here the superscript n denotes the time level Remember all solutions are periodic Let q be the steady state solution of 33 q satisfies the equation 4 F / q B / q F +/ q B +/ q + =0, =,,,M, and satisfies the constraint 4 h M q = = Condition 4 corresponds to ρxdx = We now show that q is unique, is all 0 positive, and is bounded away from 0 and from infinity, independent of the numerical grid size Theorem 4 Suppose q satisfies 4 Then q is either all zeros or all positive or all negative Proof Suppose q is not all zeros Otherwise, there is no need to continue Without loss of generality, we assume that there is an index 0 such that q 0 > 0 Otherwise, we simply consider q, which also satisfies 4 Starting at 0, we first search to the left for a nonpositive element If we cannot find a nonpositive element over a distance of M grid points, then all elements are positive because the solution is periodic Positive element Non-positive element x x x 0 x Fig 4 Starting from a positive element, q 0 > 0, we either find that all elements are positive or find a positive region bounded by nonpositive elements The latter leads to a contradiction Suppose q is the first nonpositive element found in the search to the left and q is the first nonpositive element found in the search to the right As shown in Figure 4, we have a positive region bounded by nonpositive elements: q 0, q 0, and q > 0 for <<

11 CONVERGENCE OF A NUMERICAL METHOD 435 We are going to show that this leads to a contradiction Summing 4 from = + to =, we obtain 43 B+/ q + F +/ q + F / q B / q =0 The two terms on the left-hand side of 43 are the net probability fluxes to the outside of the region Recall that the ump rates in Method are always positive Because q + > 0 and q 0, the first term is positive Similarly, the second term is also positive Thus, the total net probability flux to the outside of the region is positive This contradicts that q is a steady state Mathematically, the contradiction arises in 43, where the two positive terms sum to zero Therefore, if one element is positive, then all elements must be positive Remark The proof presented here can be extended to Fokker Planck equations of higher dimensions, in which the positive region is bounded by a combination of nonpositive elements and a periodic boundary The total net probability flux to the outside of the region is positive, which contradicts the steady state assumption Theorem 4 Suppose q satisfies 4 and 4 Then q is unique and all positive Proof Suppose both q and p satisfy 4 and 4 Let r = q p Then r satisfies 4 Applying Theorem 4 to r yields that r is either all zeros or all positive or all negative Since r also satisfies h M = r =0, r must be all zeros Consequently, q is unique Applying Theorem 4 to q and using 4, we obtain that q is all positive Remark This theorem shows that the solution of 4 with condition 4 is unique and all positive In other words, Method yields a unique steady state solution and preserves the positivity of probability As pointed above, this result can be extended to Fokker Planck equations of higher dimensions Now we consider the -norm and the weighted -norm defined as M p h p, 44 = M p ψ h p q =, h = M, where q = q, q,, q M is the solution of 4 with condition 4 In the analysis below, q =q, q,, q M is reserved to denote this steady state solution The -norm is denoted by The weighted -norm is denoted by ψ because the weighting function q depends on potential ψx Theorem 43 Suppose the free energy drop satisfies Δψ = ψx ψx +> 0 Then the steady state probability flux of 4, J = h F +/ q B +/ q +, must be positive Proof We use proof by contradiction Suppose J 0 Recall that Method satisfies detailed balance 30 It follows that q + F +/ q = e ψx ψx+ q B +/ Applying the inequality for, +,, we obtain q +M e ψx ψx +M q = e Δψ q >q,

12 436 HONGYUN WANG which contradicts the periodic condition q +M = q In the above, we have used the condition Δψ >0 Therefore, when Δψ is positive, J must be positive Remark The key component in the proof of this theorem is detailed balance 30 A method preserving detailed balance has the advantage that the direction of chemical reaction and mechanical motion is preserved in numerical results That is, the method will not produce a numerical result in which the motor system goes upward along the free energy landscape This property is very important in studies of molecular motors Theorem 44 Suppose Δψ >0, and q satisfies 4 and 4 Then we have 45 max q e C ψ, min q e C ψ, where C ψ depends on potential ψx but is independent of the numerical grid size Proof Equation 4 implies that J 46 F+/ q B +/ q + =, independent of, h where J is the steady state probability flux Applying Theorem 43, we have J >0, and 46 becomes F+/ q B +/ q + 0 Recall that Method satisfies detailed balance 30 It follows that 47 q + F +/ B +/ q = e ψx ψx+ q Let q l = min q Applying inequality 47 for = l, = l +,,weget 48 q e ψx l ψx q l for l Let C ψ = max x max x y x+ ψx ψy C ψ is a constant independent of the numerical grid size Taking the maximum of both sides of 48 over l l + M, and noticing that q is periodic, we obtain 49 max Since q also satisfies 4, we have q e C ψ q l = e C ψ min q 40 min q h M q =, = max q h M q = = Equations 49 and 40 lead immediately to 45 Remark This theorem shows that the weighted -norm is equivalent to the -norm 4 e C ψ p p ψ e C ψ p The extension of this theorem to Fokker Planck equations of higher dimensions is still an open problem we believe the theorem is valid for Fokker Planck equations of higher dimensions, but the proof is still an open problem We are going to use the weighted -norm to study the stability of the fully discrete method 3

13 CONVERGENCE OF A NUMERICAL METHOD 437 Theorem 45 Let p n =p n, p n,, p n M denote the solution of the fully discrete method 3 at time level n Then we have 4 p n+ ψ p n ψ Proof First, we write 3 as 43 p n+ p n = k h J n+/ / J n+/ +/, where 44 J n+/ +/ = h F +/ p n+/ B +/ p n+/ +, p n+/ = pn+ + p n Multiplying both sides of 43 by hp n+/, dividing by q, and summing over yields M M h p n+ h p n =k q q = = M = p n+/ J n+/ q / Applying summation by parts and using the periodic condition, we get 45 p n+ ψ p n ψ = k M = J n+/ +/ p n+/ p n+/ + J n+/ q q +/ + Let r = pn+/ q Here, for simplicity and without causing confusion, we have dropped the superscript n +/ from r We write the probability flux J n+/ +/ as J n+/ +/ = h F +/ p n+/ B +/ p n+/ + = h F +/ q r B +/ q + r + F+/ q + B +/ q + = h r r + + J r + r + Here J = h F +/ q B +/ q + is the steady state probability flux, which is a constant independent of Substituting J n+/ +/ into 45, we have 46 M p n+ ψ p n ψ = kh F+/ q + B +/ q + r r + = M k J r r + = Recall that in Method, the steady state solution q is all positive and ump rates are all positive Consequently, the first term on the right-hand side of 46 is nonpositive

14 438 HONGYUN WANG Applying summation by parts and using the periodic condition, we obtain that the second term on the right-hand side of 46 is zero Thus, we conclude that 47 p n+ ψ p n ψ 0, which immediately leads to 4 Remark This theorem shows that the fully discrete method 3 is unconditionally stable with respect to the weighted -norm ψ even if potential ψx is discontinuous Remark This theorem can be extended to Fokker Planck equations of higher dimensions 5 Consistency and convergence of the numerical method We study the consistency of Method when potential ψx is piecewise smooth and has a finite number of discontinuities in one period at half numerical grid points Without loss of generality, we assume that there is only one discontinuity at x d in [0, ], where x d = x l+/ is a half numerical grid point that is, x d is at the boundary between two numerical subintervals Below we will show that away from the discontinuity, the local truncation error of Method on the exact solution is O kk + h At the discontinuity, the local truncation error on the exact solution is O However, if we perturb the exact solution by a term of the order Oh, then the local truncation error on the perturbed solution instead of the exact solution is O kk + h 5 Local truncation error away from the discontinuity We rewrite the fully discrete method 3 in the flux form 5 p n+ = p n + k h where the numerical probability flux is J n / + J n+ / J +/ n + J n+ +/ 5 J+/ n = h F +/ p n B +/ p n + Let ρx, t be the exact solution of 4 subect to conditions 3 and 4 Let denote the exact solution on the numerical grid: ρ n ρ n = ρx,t n The local truncation error is defined as the residual term when the numerical method is applied to the exact solution ρ n Integrating the differential equation 4 over [x /,x +/ ] [t n,t n+ ], we have 53 x+/ x / ρx, t n+ dx = tn+ t n x+/ x / ρx, t n dx Jx /,tdt tn+ t n, Jx +/,tdt, where Jx, t = ψ ρ + x ρ is the exact probability flux in 4 Using the trapezoidal rule to approximate the integrals on the right-hand side yields tn+ Jx+/,t n +Jx +/,t n+ Jx +/,tdt = k t n k3 t Jx +/,t n+/ +Ok 5

15 CONVERGENCE OF A NUMERICAL METHOD 439 Since the probability flux, Jx, t, is continuous across the discontinuity, the time derivatives of Jx, t are also continuous across the discontinuity Suppose x +/ is the location of discontinuity Then we have t Jx+ +/,t n+/ = t Jx +/,t n+/ Using the assumption that everything is smooth on both sides of the discontinuity, we obtain t Jx +/,t n+/ t Jx /,t n+/ = h 3 t x Jξ,t n+/ =Oh, where x / <ξ<x +/ Expanding the integrals on the left-hand side of 53, using the results we ust derived for the integrals on the right-hand side of 53, and then dividing 53 by h, we arrive at 54 ρ n+ ρ n = k h Jx /,t n +Jx /,t n+ Jx +/,t n +Jx +/,t n+ + O kk + h Let J+/ n {ρ} denote the numerical probability flux on the exact solution ρx, t: 55 J n +/ {ρ} = h F +/ ρ n B +/ ρ n + We expand F +/, B +/, ρ n, and ρn+ around x = x +/ to obtain the following expansion for J+/ n {ρ} away from the discontinuity: 56 J n +/ {ρ} = Jx +/,t n h ux +/,t n +O h 3, where ux, t is a function consisting of various derivatives of ψx and ρx, t The derivation of 56 is presented in Appendix A Notice that ux, t is smooth in the region where ψx and ρx, t are smooth Away from the discontinuity, ux, t satisfies ux +/,t n ux /,t n =Oh Substituting 56 into 54, we obtain that, away from the discontinuity, ρ n satisfies 57 ρ n+ = ρ n + k h J n / {ρ} + J n+ / {ρ} J n n+ +/ {ρ} + J + O kk + h +/ {ρ} That is, away from the discontinuity, the local truncation error on the exact solution ρx, t iso kk + h

16 440 HONGYUN WANG ψ R ψ l+ ψx ψ l ψ L x l x l+/ x d x l+ x Fig 5 Schematic diagram of the discontinuity at x d = x l+/ 5 Local truncation error on the perturbed solution Now let us look at the numerical probability flux at the discontinuity As shown in Figure 5, the discontinuity is at x d = x l+/ We introduce several shorthand notations: ρ n L = ρx, t n x=x, ρ n R = ρx, t n d x=x +, d ψ L = ψx x=x, ψ R = ψx d x=x + d The numerical probability flux on the exact solution ρx, t is Jl+/ n {ρ} = h F l+/ ρ n l B l+/ ρ n l+ = ψ l+ ψ l h e ψ l+ e ψ e ψ l ρ n l l e ψ l+ ρ n l+ = ψ l+ ψ l h e ψ l+ e ψ e ψ l ρ n l l e ψ L ρ n L + e ψ R ρ n R e ψ l+ ρ n 58 l+ Here we have used condition 4: e ψ L ρ n L = eψ R ρ n R Expanding eψ l ρ n l and e ψ l+ ρ n l+ around x = x+ d,weget around x = x d e ψ l ρ n l e ψ L ρ n L = h eψ L ψ xρ + ρ x x=x d + h v L t n +h 3 w L t n +Oh 4 = h eψ L Jx l+/,t n +h v L t n +h 3 w L t n +Oh 4, e ψ R ρ n R e ψ l+ ρ n l+ = h eψ R Jx l+/,t n +h v R t n +h 3 w R t n +Oh 4, where v L t, v R t, w L t, and w R t are smooth functions of t Substituting these two expansions into 58, we have 59 J n l+/ {ρ} = Jx l+/,t n +v 0 t n +hv t n +h v t n +Oh 3,

17 CONVERGENCE OF A NUMERICAL METHOD 44 where [ ψr ψ L e ψ L + e ψ R 50 v 0 t = e ψ R e ψ L ] Jx l+/,t n Again, v 0 t, v t, and v t are smooth functions of t Combining 59, which gives numerical flux at the discontinuity, and 56, which is valid away from the discontinuity, we obtain 5 where 5 53 J / n {ρ} J +/ n {ρ} = Jx /,t n Jx +/,t n + Oh 3 + h ux + d,t n ux d,t n a + h v 0 t n +hv t n +h ṽ t n b, a =, = l,, = l +, h M M otherwise, h, = l, b = + h, = l +, 0 otherwise, and 54 ṽ t =v t+ ux+ d,t+ux d,t In 5, the first term on the right-hand side is the desired result; the second term is of the order Oh 3 ; the third term is of the order Oh at the discontinuity and of the order Oh 3 elsewhere; the fourth term is of the order O, which is the term we need to deal with Terms in 5 contribute to the local truncation error In general, the global error is one order lower than the local truncation error However, the O term in 5 does not necessarily imply that the error in the numerical solution is O/h or is O As we will see below, the O term in 5 actually leads to an Oh term in the global error The key is that the O term in 5 can be eliminated by perturbing the exact solution by an Oh term So if we use the perturbed exact solution to calculate the local truncation error, then the O term in 5 disappears In other words, the numerical method is consistent with the perturbed exact solution and the perturbed exact solution converges to the exact solution The successful elimination of the O term in 5 by perturbing the exact solution by an Oh term depends on the special structure of vector {b } given in 53 For that purpose, we have the theorem below Theorem 5 Suppose r is periodic, satisfies the equation 55 h F / r B / r h F+/ r B +/ r + = b, and satisfies the condition 56 M r =0, =

18 44 HONGYUN WANG where b is defined in 53 Then there exists a constant C b, independent of the numerical grid size, such that max r C b Proof The proof of Theorem 5 is presented in Appendix B Now we use the result of Theorem 5 to eliminate the O error term in 5, and we consider the perturbed solution given below: 57 ρ n ρ n h v 0 t n +hv t n +h ṽ t n r = ρ n + Oh, where r is the solution of 55 and 56 in Theorem 5 It is straightforward to verify that the perturbed solution ρ n satisfies J / n { ρ} J +/ n { ρ} = Jx /,t n Jx +/,t n + Oh, ρ n+ ρ n = ρ n+ ρ n + Okh Substituting 58 and 59 into 54, we obtain 50 ρ n+ = ρ n + k h J n / { ρ} + J n+ / { ρ} J n n+ +/ { ρ} + J + O kk + h +/ { ρ} That is, the local truncation error on the perturbed solution ρ n is O kk + h 53 Convergence of the numerical method Once we have both the stability and the consistency, the convergence follows, in principle, from the Lax equivalence theorem [7] More specifically, we write the numerical method 3 in the vectoroperator form 5 p n+ = L p n, where L is the linear operator representing the numerical method Stability 4 implies 5 L ψ, where L ψ is the induced operator norm defined by L ψ max p ψ = L p ψ Consistency 50 on the perturbed solution implies 53 ρ n+ = L ρ n + O kk + h, where ρ n is the perturbed solution given in 57 Subtracting 53 from 5 yields [ 54 p n+ ρ n+ = L p n ρ n] + O kk + h

19 CONVERGENCE OF A NUMERICAL METHOD 443 Taking ψ norm of the both sides, using the stability, and summing over n, we have 55 p n ρ n ψ T Ok + h, nk T Using the fact that ρ n ρ n = Oh, we obtain 56 p n ρ n ψ T Ok + h Using Theorem 44, we see that 56 implies the convergence in the -norm: 57 p n ρ n e C ψ p n ρ n ψ e C ψ T Ok + h If k Oh, then 56 also implies the pointwise convergence 58 p n ρ n h p n ρ n e C ψ T O h 6 The modified numerical method In the consistency analysis of the previous section, we see the connection between the leading term in the local truncation error and the ump rates Based on the lessons we learned in the consistency analysis, we will design a new set of ump rates to eliminate the first order error term caused by the discontinuity The modified numerical method hereafter referred to as Method is as simple as Method We will show that Method is second order accurate even if the potential is discontinuous In the local truncation error in 50, the leading term Okh comes from the term v 0 t n in 59 At the discontinuity, if we use Method defined in 39, then we have [ ψr ψ L e ψ L + e ψ ] R v 0 t n = e ψ R e ψ L Jx l+/,t n 0 This suggests a way of improving the performance of Method We need to design the ump rates such that v 0 t n = 0 at the discontinuity For that purpose, we propose Method : F +/ = e ψ h e ψ + e, ψ+ B +/ = e ψ+ 6 h e ψ + e ψ+ If we use Method defined in 6, then we have [ e ψ L + e ψ ] R v 0 t n = e ψ L + e ψ R Jx l+/,t n =0 It can be shown that expansion 56 is still valid for Method Consequently, expansion 5 is valid for Method where v 0 t n = 0 Recall that for Method, the O term in 5 leads to an Oh error term in the numerical solution For Method, v 0 t n = 0 kills the O term in 5 The third term in 5 is Oh, and the remaining part of the fourth term in 5 is Oh Because of the special structures of vector {b } in 53 and vector {a } in 5, both the third term and the remaining part of the fourth term in 5 can be eliminated by perturbing the

20 444 HONGYUN WANG exact solution by an Oh term The elimination of these two terms by a perturbation of Oh makes Method a second order method For eliminating the third term in 5, we have the theorem below Theorem 6 Suppose σ is periodic, satisfies the equation 6 h F / σ B / σ h F+/ σ B +/ σ + = a, and satisfies the condition 63 M σ =0, = where a is defined in 5 Then there exists a constant C a, independent of the numerical grid size, such that max σ C a Proof The proof of Theorem 6 is similar to that of Theorem 5 and is skipped Now we use the results of Theorems 5 and 6 to eliminate the third and fourth terms in 5 For Method, we consider the perturbed solution: 64 ˆρ n ρ n h v t n +h ṽ t n r h ux + d,t n ux d,t n σ = ρ n + Oh, where r is the solution of 55 and 56, and σ is the solution of 6 and 63 Theorems 5 and 6 guarantee that both r and σ are bounded by a constant, independent of the numerical grid size The perturbed solution ˆρ n satisfies 65 J / n {ˆρ} J +/ n {ˆρ} = Jx /,t n Jx +/,t n + Oh 3, 66 ˆρ n+ ˆρ n = ρ n+ ρ n + Okh Thus, the local truncation error on the perturbed solution ˆρ n is O kk + h Repeating the derivation from 5 to 58, we obtain p n ρ n e C ψ T Ok + h, p n ρ n e C ψ T Oh 3 The error bound for the -norm 68 is derived assuming the worst case scenario As we will see in the numerical example below, the -norm of the error is usually of the same order as the -norm of the error Method defined in 6 can be viewed as constructed by using the standard finite difference on with a special approximation for the heat conductivity: e ψx+/ e ψx + e ψx+ This special approximation is essential for Method to achieve second order accuracy at discontinuities Method developed in [6] can be viewed as constructed by using the standard finite difference on with approximation e ψx +/ ψx + ψx e ψx+ e ψx

21 CONVERGENCE OF A NUMERICAL METHOD 445 The numerical method previously developed by Elston and Doering in [8] can be viewed as constructed by using the standard finite difference on with approximation e ψx+/ e ψx +ψx + 7 Numerical results and discussions Now we compare the performance of three numerical methods on a model problem with discontinuous potential For the model problem, we select the potential { 6 6 sin π 7 ψx = x +05, 0 <x<05, 3 6 sin π x 05, 05 <x< The graph of this discontinuous potential is shown in Figure It has a discontinuity of amplitude 3 at x =05 We use the initial condition 7 ρx, 0 = + cosπx We run simulations to t = with a wide range of values for spatial step h, and we use time step k = h We compare the performance of the central difference method 34, Method 39, and Method 6 We define the error as the difference between the numerical solution obtained with a finite value of h and the converged target the converged target is not necessarily the correct solution of the differential equation The behavior of the error so defined tells us whether or not a method converges However, it does not tell us whether or not the converged target is the correct solution We estimate the error as follows Suppose p n h is the numerical solution obtained with spatial step h and time step k = h The error of p n h is estimated as 73 errorh C p p n h p n h, where C p is a constant depending on the order of the method For methods of first order or higher, C p is between and Here we simply use C p = in the worst case scenario, we underestimate the error by a factor of The order of a method is estimated as errorh 74 orderh log error h Figure 7 shows the estimated errors and estimated orders for the three methods We follow the behavior of both the -norm and the -norm of the estimated error Here we are solving a nondimensionalized Fokker Planck equation Both the time step and the spatial step are dimensionless So k = h is ust a convenient choice Strictly speaking, the error shown in Figure 7 is the total error in time and space estimated by comparing the numerical solution obtained using h, k to that of h, k However, we find that the difference in numerical solution between h, k and h, k is much smaller than that between h, k and h, k results not shown, which indicates that the error shown in Figure 7 is mainly due to the spatial discretization For Method the second row in Figure 7, the estimated error decreases as k = h is reduced From the convergence analysis in the previous sections, we know that

22 446 HONGYUN WANG 0 0 Central difference Central difference errorh 0 - orderh norm -norm h Method 0 -norm -norm 0-3 h 0 - Method errorh norm -norm h 0 - orderh -norm -norm Method Method h errorh norm -norm h 0 - orderh -norm -norm h 0 - Fig 7 Estimated errors first column and estimated orders second column for the central difference method first row, Method second row, and Method third row the converged target must be the correct solution of the Fokker Planck equation In the presence of a discontinuity, the estimated order of accuracy of Method developed in [6] is for both the -norm and the -norm This result is consistent with the error bounds 57 and 58 derived in the previous sections Notice that the -norm and the -norm of the estimated error are of the same order This tells us that although the first order error is caused by the discontinuity, it is spread to the whole region by diffusion For Method the third row in Figure 7, the estimated error decreases more rapidly as k = h is reduced The convergence analysis in the previous sections guarantees that Method converges to the correct solution of the Fokker Planck equation

23 CONVERGENCE OF A NUMERICAL METHOD Central difference Method Method 8 ρx x Fig 7 Numerical probability densities at t =obtained with spatial and time steps h = k = using, respectively, the central difference method, Method, and Method 4096 Even in the presence of a discontinuity, the estimated order of accuracy for Method is for both the -norm and the -norm This result is consistent with the error bounds 67 and 68 derived in the previous section For the central difference method the first row in Figure 7, it appears that the estimated error converges to zero as k = h goes to zero The convergence to the target is very slow Even with k = h = 4096, both the -norm and the -norm of the error are still above 0 But this is not the only defect of the central difference method The fatal defect of the central difference method is that it converges to a wrong solution that does not satisfies condition 4 at the discontinuity Figure 7 shows the numerical probability densities at t = for the three methods Notice that the probability density obtained using the central difference method is negative for x>05 This is definitely wrong because the probability can never be negative This defect of the central difference method is caused by the fact that the ump rates 34 may be negative at the discontinuity Suppose the discontinuity is at x l+/ The numerical probability flux at x l+/ is J l+/ = h F l+/ p l B l+/ p l+ 75 = hb l+/ p l [ Fl+/ B l+/ p l+ p l A necessary condition for converging to the correct solution is Applying condition 4 yields p l+ lim h 0 p l lim h 0 p l = ρx d,t, lim h 0 p l+ = ρx + d,t, lim h 0 J l+/ = finite = ρx+ d,t ρx d = eψx,t ] d ψx+ d

24 448 HONGYUN WANG Multiplying 75 by h, using h B l+/ = O, and taking the limit as h 0, we obtain F l+/ lim = e ψx d ψx+ d, h 0 B l+/ which reduces to detailed balance 30 Therefore, in the presence of discontinuities, detailed balance is a necessary condition for converging to the correct solution of the differential equation Both Method developed in [6] and Method proposed in this paper satisfy detailed balance The central difference method does not For the model problem 7, we have e ψx d ψx+ d = e 3, F CD l+/ lim h 0 B CD l+/ = ψx + d ψx d + ψx+ d ψx d = 5 < 0 The negative value of F l+/ B l+/ will force the ratio p l+ p l to be negative, which leads to negative probability in the numerical solution of the central difference method In conclusion, we have proved that Method developed in [6] is stable and is consistent with the Fokker Planck equation Method converges to the correct solution of the Fokker Planck equation, and the -norm of the error behaves like Ok + h when the potential is discontinuous Numerical results indicate that the -norm of the error is of the same order Based on the consistency analysis, we proposed a modified version of Method to eliminate the first order error caused by the discontinuity The modified numerical method Method is guaranteed to converge to the correct solution, and the -norm of the error behaves like Ok + h even in the presence of discontinuities Again, numerical results indicate that the -norm of the error is of the same order In stochastic ratchets with discontinuous force [0], also known as sharp stochastic ratchets [], the potential is continuous, but the derivative of the potential is discontinuous Originally in [0] and subsequently in [], the transport in stochastic ratchets was studied where a particle is driven by a continuous piecewise linear potential, the Brownian noise white noise, and an additional colored noise They derived analytic expressions for the steady state particle current for various asymptotic limits Now we look at the convergence of numerical methods in the case of sharp stochastic ratchets When the potential is continuous and piecewise smooth, both Method and Method the modified numerical method converge, and the error behaves like Ok + h This can be seen by going back to section 5 In the local truncation error in 50, the leading term Okh comes from the term v 0 t n in 59 v 0 t n is nonzero only at discontinuities For Method, v 0 t n is given in 50 At a discontinuity on the derivative of a continuous function, we have ψ L = ψ R and consequently v 0 t n = 0 Thus, Method is second order when the potential is continuous Method the modified numerical method is already second order even when the potential is discontinuous All of the conclusions above for Method are also true for the numerical method previously developed by Elston and Doering [8] More specifically, the numerical stability proved in section 4 depends on two main features of the numerical method, i all ump rates being positive and ii detailed balance being preserved, which are satisfied in the method of [8] The numerical consistence away from the discontinuities is obtained by doing Taylor expansion The key feature we utilized in section 5