A Dynamically Bi-Orthogonal Method for Time-Dependent Stochastic Partial Differential Equations I: Derivation and Algorithms

Transcription

1 A Dynamically Bi-Orthogonal Method for ime-dependent Stochastic Partial Differential Equations I: Derivation and Algorithms Mulin Cheng a, homas Y. Hou a,, Zhiwen Zhang a a Computing and Mathematical Sciences, California Institute of echnology, Pasadena, CA 925 Abstract We propose a dynamically bi-orthogonal method DyBO to solve time dependent stochastic partial differential equations SPDEs. he objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a biorthogonal basis. It is well-known that the Karhunen-Loeve expansion KLE minimizes the total mean squared error and gives the sparsest representation of stochastic solutions. However, the computation of the KL expansion could be quite expensive since we need to form a covariance matrix and solve a large-scale eigenvalue problem. he main contribution of this paper is that we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the KL expansion. nlike other reduced model methods, our method constructs the reduced basis on-the-fly without the need to form the covariance matrix or to compute its eigendecomposition. In the first part of our paper, we introduce the derivation of the dynamically bi-orthogonal formulation for SPDEs, discuss several theoretical issues, such as the dynamic bi-orthogonality preservation and some preliminary error analysis of the DyBO method. We also give some numerical implementation details of the DyBO methods, including the representation of stochastic basis and techniques to deal with eigenvalue crossing. In the second part of our paper [], we will present an adaptive strategy to dynamically remove or add modes, perform a detailed complexity analysis, and discuss various generalizations of this approach. An extensive range of numerical experiments will be provided in both parts to demonstrate the effectiveness of the DyBO method. Keywords: Stochastic partial differential equations, Karhunen-Loeve expansion, uncertainty quantification, bi-orthogonality, reduced-order model.. Introduction ncertainty arises in many complex real-world problems of physical and engineering interests, such as wave, heat and pollution propagation through random media [3, 9, 7, 53] and flow driven by stochastic forces [26, 28, 5, 42, 37]. Additional examples can be found in other branches of science and engineering, such as geosciences, statistical mechanics, meteorology, biology, finance, and social science. Stochastic partial differential equations SPDEs have played an important role in our investigation of uncertainty quantification Q. However, it is very challenging to solve stochastic PDEs efficiently due to the large dimensionality of stochastic solutions. Although many numerical methods have been proposed in the literature see e.g. [52, 9, 2, 3, 23, 3, 38, 54, 4, 5, 55, 36, 5, 28, 48,, 39, 24, 46, 4, 34, 5, 6, 4, 44], it is still very expensive to solve a time dependent nonlinear stochastic PDEs with high dimensional stochastic input variables. his limits our ability to attack some challenging real-world applications. In this paper, we propose a dynamically bi-orthogonal method DyBO to solve time-dependent stochastic partial differential equations. he dynamic bi-orthogonality condition that we impose on DyBO is closely Corresponding author addresses: mulinch@caltech.edu Mulin Cheng, hou@cms.caltech.edu homas Y. Hou, zhangzw@caltech.edu Zhiwen Zhang Preprint submitted to Elsevier Friday 8 th February, 23

2 related to the Karhunen-Loeve expansion KLE [29, 32]. he KL expansion has received increasing attention in many fields, such as statistics, image processing, and uncertainty quantification. It yields the best basis in the sense that it minimizes the total mean squared error and gives the sparsest representation of stochastic solutions. However, computing KLE could be quite expensive since it involves forming a covariance matrix and solving the associated large-scale eigenvalue problem. One of the attractive features of our DyBO method is that we construct the sparsest bi-orthogonal basis on-the-fly without the need to form the covariance matrix or to compute its eigendecomposition. We consider the following time-dependent stochastic partial differential equations: u x, t, ω = Lux, t, ω, x D Rd, ω Ω, t [, ], a ux,, ω = u x, ω, x D, ω Ω, b Bux, t, ω = hx, t, ω, x D, ω Ω, c where L is a differential operator and may contain random coefficients and/or stochastic forces and B is a boundary operator. he randomness may also enter the system through the initial condition u and/or the boundary conditions B. We assume the stochastic solution ux, t, ω of the system is a second-order stochastic process, i.e., u, t, L 2 D Ω. he solution ux, t, ω can be represented by its KL expansion as follows: ux, t, ω = ūx, t + u i x, ty i ω, t, 2 where ūx, t = E [ux, t, ω], u i s are the eigenfunctions of the associated covariance function Cov u x, y = E [ux, t, ω ūx, tuy, t, ω ūy, t]. hus u i are orthogonal to each other, i.e., u i, u j = λ i δ ij, where λ i are the corresponding eigenvalues of the covariance function. he random variables Y i have zero-mean and are mutually orthogonal, i.e. E [Y i Y j ] = δ ij. hey are defined as follows: Y i ω, t = λ i t D i= ux, t, ω ūx, tu i x, t dx, for i =, 2, 3,. For many physical and engineering problems, the dimension of the input stochastic variables may appear to be high, but the effective dimension of the stochastic solution may be low due to the rapid decay of the eigenvalues λ i [48]. For this type of problems, the KLE provides the sparest representation of the solution through an optimal set of bi-orthogonal spatial and stochastic basis. Since this bi-orthogonal basis is constructed at each time, both the spatial basis u i and the stochastic basis Y i are time-dependent to preserve bi-orthogonality. his is very different from other reduced-order basis in which either the spatial or the random basis is time-dependent, but not both of them are time-dependent. he traditional way to construct the KL expansion is to first solve SPDE by some conventional method, and then form the covariance function to construct the KL expansion. As we mentioned earlier, this post-processing approach is very expensive which would offset the benefit of giving a sparsest representation of the stochastic solution. he main objective of this paper is to derive a well-posed system that governs the dynamic evolution of the KL expansion without the need of forming the covariance function explicitly. o illustrate the main idea, we assume that the eigenvalues λ i decay rapidly and an m-term truncation in the KL expansion would give an accurate approximation of the stochastic solution. We substitute the m-term truncated expansion 2, denoted as ũ, into the stochastic PDE a. After projecting the resulting equation into the spatial and stochastic basis and using the bi-orthogonality of the basis, we obtain = E [Lũ], = Λ = YE, [ Y 3a + G ū,, Y, 3b ] + G Y ū,, Y, 3c 2

3 where E [ ] is the expectation, f, g = fxgxdx, x, t = u x, t, u 2 x, t,..., u m x, t, Yω, t = Y ω, t, Y 2 ω, t,..., Y m ω, t, ũ = ū + Y, Λ = diag,, G ū,, Y and G Y ū,, Y are defined in 2a and 2b respectively. he initial condition of 3 is given by the KL expansion of u x, ω. A close inspection shows that the evolution system 3b and 3c does not determine the time derivative or uniquely. In fact, 3b only determines the projection of into the orthogonal complement set of uniquely, which is G. Similarly, 3c only determines the projection of into the orthogonal complement set of Y uniquely, which is G Y. hus, the evolution system 3 can be rewritten as follows: = E [Lũ], = C + G ū,, Y, = YD + G Yū,, Y, where C and D are m m matrices which represent the projection coefficients of and into and Y respectively. hese two matrices characterize the large degrees of freedom in choosing the spatial and stochastic basis dynamically. hey cannot be chosen independently. Instead, they must satisfy a compatibility condition to be consistent with the original SPDE and satisfy the bi-orthogonality condition of the spatial and the stochastic basis. By introducing two types of antisymmetric operators to enforce the bi-orthogonality condition, we derive two constraints for C and D. When combining these two constraints with the compatibility condition, we show that these two matrices can be determined uniquely provided that the eigenvalues of the covariance matrix are distinct. We prove rigorously that the system of governing equations indeed preserve the bi-orthogonality condition dynamically. More precisely, if we write χt = u i, u j i>j as a column vector in R mm 2, then χt satisfies a linear ODE system dχt = Wtχt for some anti-symmetric matrix Wt. he fact that Wt is anti-symmetric is important because this implies that the dynamic preservation of bi-orthogonality condition is very stable to perturbation and any deviation from the bi-orthogonal form will not be amplified. he breakdown in deriving a well-posed system for the KL expansion when two distinct eigenvalues coincide dynamically is associated with additional degrees of freedom in choosing orthogonal eigenvectors. o overcome this difficulty, we design a strategy to handle eigenvalue crossings. When two distinct eigenvalues become close, we temporarily freeze the stochastic basis and evolve only the spatial basis. By doing so, we still maintain the dynamic orthogonality of the stochastic basis, which allows us to compute the covariance matrix relatively easily. After the two eigenvalues separate in time, we can reinitialize the DyBO method by reconstructing the KL expansion without forming the covariance matrix. Similarly, we can freeze the spatial basis and evolve the stochastic basis for a short time until the two eigenvalues are separated. We remark that the low-dimensional or sparse structures of SPDE solutions have been explored partially in some methods, such as the proper orthogonal decomposition POD methods [3, 49, 5], reduced-basis RB methods [7, 35, 45, 2], and dynamical orthogonal method DO [46, 47]. Our method shares some common feature with the dynamically orthogonal method since both methods use time-dependent spatial and stochastic basis. An important ingredient in POD and RB methods is how to choose the reduced basis from some limited information of the solution that we learn offline. he main difference between our method and other reduced basis methods is that we bypass the step of learning the basis offline, which in general can be quite expensive. We construct the reduced basis on the fly as we solve the SPDE. We have performed some preliminary error analysis for our method. In the case when the differential operator is linear and deterministic and the randomness enters through the initial condition or forcing, we show that the error is due to the projection of the initial condition or the random forcing into the biorthogonal basis. In the special case when the initial condition has exactly m modes in the KL expansion, the m-term DyBO method will reproduce the exact solution without any error. For general nonlinear stochastic PDEs, the number of effective modes in the KL expansion will increase dynamically due to nonlinear interaction. In this case, even if the initial condition has an exactly m-mode KL expansion, we must dynamically increase modes to maintain the accuracy of our computation. 3 4a 4b 4c

4 λi DyBO wi L 2 gpc wα L 2 gpc DyBO 2 2 e.62m e.65np Figure : Stochastically forced Burgers equations computed by DyBO and gpc. Left: Sorted Energy spectrum of gpc and DyBO solutions. Right: Relative errors of SD, DyBO e.62m vs gpc e.65np. We have developed an effective adaptive strategy for DyBO to add or remove modes dynamically to maintain a desirable accuracy. If the magnitude of the last mode falls below a prescribed tolerance, we can remove this mode dynamically. On the other hand, when the magnitude of the last mode rises above a certain tolerance, we need to add a new mode pair. he method we propose to add new modes dynamically is to use the DyBO solution as the initial condition and compute a DyBO solution and a generalized Polynomial Chaos gpc solution respectively for a short time. We then compute the KL expansion of the difference between the DyBO solution and the gpc solution. We add the largest mode from the KL expansion of the difference to our DyBO method. his method works quite effectively. More details will be reported in the second part of our paper []. As we will demonstrate in our paper, the DyBO method could offer accurate numerical solutions to SPDEs with significant computational saving over traditional stochastic methods. We have performed numerical experiments for D Burgers equations, 2D incompressible Navier-Stokes equations, and Boussinesq equations with approximated Brownian motion forcing. In all these cases, we observe considerable computational saving. he specific rate of saving will depend on how we discretize the stochastic ordinary differential equations SODEs governing the stochastic basis. We can use three methods to discretize the SODEs, i.e. i Monte Carlo methods MC, ii generalized stochastic collocation methods gsc, iii generalized polynomial chaos methods gpc. In the numerical experiments we present in this paper, we use gpc to discretize the SODE. Our numerical results show that the DyBO method gives much faster decay rate for its error compared with that of the gpc method. o illustrate this point, we show the sorted energy spectrum of the solution of the stochastically forced Burgers equation computed by our DyBO method and that of the gpc method respectively in Figure. We can see that the eigenvalues of the covariance matrix decay much faster than the corresponding sorted energy spectrum of the gpc solution, indicating the effective dimension reduction induced by the bi-orthogonal basis. If we denote m as the number of modes used by DyBO and N p the number of modes used by gpc, our complexity analysis shows that the ratio between the computational complexities of DyBO and gpc is roughly of order Om/N p 3 for a two-dimensional quadratic nonlinear SPDE. ypically m is much smaller than N p, as illustrated in Figure. More details can be found in part II of our paper where we also discuss the parallel implementation of DyBO []. his paper is organized as follows. In Section 2, we provide the detailed derivation of the DyBO formulation for a time-dependent SPDE. Some important theoretic aspects of the DyBO formulation, such as bi-orthogonality preservation and the consistency between the DyBO formulation and the original SPDE are discussed in Section 3. In Section 4, we discuss some detailed numerical implementations of the DyBO method, such as the representation of the stochastic basis Y and a strategy to deal with eigenvalue crossings. hree numerical examples are provided in Section 5 to demonstrate computational advantage of our DyBO method. Finally, some remarks will be made in Section 6. 4

5 2. Derivation of dynamically bi-orthogonal formulation In this section, we introduce the derivation of our dynamically bi-orthogonal method. For the ease of future derivations, we use extensively vector and matrix notations. x, t = u x, t, u 2 x, t,..., u m x, t, Ũx, t = u m+ x, t, u m+2 x, t,..., Yω, t = Y ω, t, Y 2 ω, t,..., Y m ω, t, Ỹω, t = Y m+ ω, t, Y m+2 ω, t,.... Correspondingly, E[Y Y] and, are m-by-m matrices and the bi-orthogonality condition can be rewritten compactly as E [ Y Y ] t = E [Y i Y j ] = I R m m, 5a, t = u i, u j = Λ R m m, 5b where Λ = diag, = λ i δ ij m m. herefore, the KL expansion 2 of u at some fixed time t reads u = ū + Y + ŨỸ. 6 Furthermore, we assume that the solution of the SPDE enjoys a low-dimensional structure in the sense of KLE, or precisely, the eigenvalue spectrum {λ i t} i= decays fast enough, allowing for a good approximation with a truncated KL expansion 2. We denote by ũ as the m-term truncation of the KL expansion of u, ũ = ū + Y. 7 We introduce the following orthogonal complementary operators with respect to the orthogonal basis in L 2 D or Y in L 2 Ω to simplify notations, Π v = v Λ, v for vx L 2 D, Π Y Z = Z YE [ Y Z ] for Zω L 2 Ω. We also define an anti-symmetrization operator Q : R k k R k k and a partial anti-symmetrization operator Q : R k k R k k, which are defined as follows: QA = 2 A A, QA = A A + diaga, 2 where diaga denotes a diagonal matrix whose diagonal entries are equal to those of matrix A. In the DyBO formulation it is crucial to dynamically preserve the bi-orthogonality of the spatial basis and the stochastic basis Y. We begin the derivation by substituting the KLE 2 into the SPDE a and get + Y + = Lũ + {Lu Lũ} { } Ũ Ỹ + Ũ dỹ. If the eigenvalues in the KL expansion decay fast enough and the differential operator is stable, the last two terms on the right hand side will be small, so we can drop them and obtain the starting point for deriving our DyBO method, + Y + = Lũ. 9 aking expectations on both sides of Eq. 9 and taking into account the fact that Y i s are zero-mean random variables, we have = E [Lũ], 5

6 which gives the evolution equation for the mean of the solution u. Multiplying both sides of Eq. 9 by Y from the right and taking expectations, we obtain E [ Y Y ] [ ] [ + E Y = E Lũ ] Y. After using the orthogonality of the stochastic basis Y, we have [ ] [ ] = E LũY E Y, where Lũ = Lũ E [Lũ]. Similarly we obtain an evolution equation for. herefore, this gives rise to the following evolution system: = E [Lũ], a [ ] [ ] = E LũY E Y, b Λ = Lũ, Y,. c However, the above system does not give explicit evolution equations for the evolution system, we substitute Eq. c into Eq. b and get = E [ LũY ] [ ] Λ, E LũY + Λ and. o further simplify,, where we have used the orthogonality of the stochastic basis, i.e., E [ Y Y ] = I. Similar steps can be repeated by substituting Eq. b into Eq. c. hus, the system can be re-written as = E [Lũ], = Λ = YE, [ Y a + G ū,, Y, b ] + G Y ū,, Y, c where we have used the orthogonality condition of the spatial and stochastic modes, and [ ] [ ] [ ] G ū,, Y = Π E LũY = E LũY Λ, E LũY, 2a [ G Y ū,, Y = Π Y Lũ, Λ = Lũ, Λ E ] Y LũY, Λ. 2b [ ] Note that G is the projection of E LũY into the orthogonal complementary set of and G Y is the projection of Lũ, into the orthogonal complementary set of Y. hus, we have G ū,, Y span in L 2 D, G Y ū,, Y spany in L 2 Ω. he above observation will be crucial in the later derivations. We note that the system b and c does not determine the time derivative In fact, if is a solution of b, then uniquely. + C for any m m matrix C is also a solution of b. or 6

7 Similar statement can be made for. In Appendix A, we will show that the evolution system 3 can be rewritten as follows: = E [Lũ], = C + G ū,, Y, = YD + G Yū,, Y, where C and D are m m matrices which represent the projection coefficients of and into and Y respectively. Moreover, we show that by enforcing the bi-orthogonality condition of the spatial and stochastic basis and a compatibility condition with the original SPDE, these two matrices can be determined uniquely provided that the eigenvalues of the covariance matrix are distinct. he results are summarized in the following theorem. heorem 2. Solvability of matrices C and D. If u i, u i u j, u j for i j, i, j =, 2,..., m, the m-by-m matrices C and D can be solved uniquely from the following linear system 3a 3b 3c C Λ Λ C =, 4a D Q D =, 4b D + C = G ū,, Y, [ ] where G ū,, Y = Λ, E LũY R m m. he solutions are given entry-wisely as follows: 4c C ii = G ii, 5a u j 2 L C ij = 2 D u j 2 L 2 D u i 2 G ij + G ji, for i j, 5b L 2 D D ii =, D ij = u j 2 L 2 D u i 2 L 2 D 5c u j 2 L 2 D G ji + u i 2 L 2 D G ij, for i j. 5d o summarize the above discussion, the DyBO formulation of SPDE results in a set of explicit equations for all the unknowns quantities. In particular, we reformulate the original SPDE into a system of m stochastic ordinary differential equations for the stochastic basis Y coupled with m + deterministic PDEs for the mean solution ū and spatial basis. Further, from the definition of G given in heorem 2., we can rewrite G and G Y as [ ] G = E LũY G, G Y = Lũ, Λ YG. From the above identities, we can rewrite the DyBO formulation 3 as follows: = E [Lũ], 6a [ ] = D + E LũY, 6b = YC + Lũ, Λ, 6c where we have used the compatibility condition Eq.4c for C and D, i.e., D + C = G. he initial conditions of ū, and Y are given by the KL expansion of u x, ω in Eq.b. We can derive the 7

8 boundary condition of ū and the spatial basis by taking expectation on the boundary condition. For example, if B is a linear deterministic differential operator, we have Būx, t, ω x D = E [hx, t, ω] and Bu i x, t, ω x D = E [hx, t, ωy i ω, t]. If the periodic boundary condition is used in Eq.c, ū and the spatial basis satisfy the periodic boundary condition. Remark 2.. In the event that two eigenvalues approach each other and then separate as time increases, we refer to this as eigenvalue crossings. he breakdown in determining matrices C and D when we have eigenvalue crossings represents the extra degrees of freedom in choosing a complete set of orthogonal eigenvectors for repeated eigenvalues. o overcome this difficulty associated with eigenvalue crossings, we can detect such an event and temporarily freeze the spatial basis or the stochastic basis Y and continue to evolve the system for a short duration. Once the two eigenvalues separate, the solution can be recast into the bi-orthogonal form via KL expansion. Recall that one of and Y is always kept orthogonal even during the freezing stage. We can devise a KL expansion algorithm which avoids the need to form the covariance function explicitly. More details will be discussed in the section about numerical implementation. 3. heoretical analysis 3.. Bi-Orthogonality Preservation In the previous section, we have used the bi-orthogonal condition in deriving our DyBO method. Since we have made a number of approximations by using a finite truncation of the KL expansion and performing various projections, these formal derivations do not necessarily guarantee that the DyBO method actually preserves the bi-orthogonal condition. In this section, we will show that the DyBO formulation 6 preserves the bi-orthogonality of the spatial basis and the stochastic basis Y for t [, ] if the bi-orthogonality condition is satisfied initially. heorem 3. Preservation of Bi-Orthogonality in DyBO formulation. Assume that there is no eigenvalue crossing up to time >, i.e. u j L2 D u i L2 D for i j. he solutions and Y of system 6 satisfy the bi-orthogonality condition 5 exactly as long as the initial conditions x, and Yω, satisfy the bi-orthogonality condition 5. Moreover, if we write χ i,j t = u i, u j t for i > j and denote χt = χ i,j t mm as a column vector in R 2, then there exists an anti-symmetric matrix Wt i>j R mm 2 mm 2 such that Similar results also hold for the stochastic basis Y. dχt = Wtχt, 7a χ =. 7b Remark 3.. he fact that Wt is anti-symmetric is important for the stability of our DyBO method in preserving the dynamic bi-orthogonality of the spatial and stochastic basis. Due to numerical round-off errors, discretization errors, etc, and Y may not be perfectly bi-orthogonal at the beginning or become so later in the computation. he above theorem sheds some lights on the numerical stability of DyBO formulation in this scenario. Since the eigenvalues of an anti-symmetric matrix are purely imaginary or zero and their geometric multiplicities are always one see page 5, [22], so any deviation from the bi-orthogonal form will not be amplified. Proof of heorem 3.. We only give the proof for the spatial basis, since the proof for the stochastic basis Y is similar. Direct computations give d, =, +, = D, [ ] + E LũY,, [ ] D +, E LũY = D,, D + G Λ + Λ G, 8 8

9 where we have used the definition of G see heorem 2. and A.3 in the last equality. For each entry with i j in 8, we get, d u i, u j = = = m m D ik u k, u j u i, u l D jl + G ji u j 2 L 2 D + G ij u i 2 L 2 D k= m k=,k j l= D ik u k, u j m l=,l i u i, u l D jl D ij u j 2 L 2 D D ji u i 2 L 2 D u + j 2 L 2 D u i 2 L 2 D D ij m m D ik u k, u j D jl u i, u l, 9 k=,k i,j l=,l i,j where Eq. 5d and the anti-symmetric property of matrix D are used. From Eq. 9, we see that χt satisfies a linear ODE system in the form of 7 with matrix Wt given in terms of D ij and the initial condition comes from the fact that x, are a set of orthogonal basis. It is easy to see that, as long as the solutions of system 6 exist, Wt is well-defined and the ODE system admits only zero solution, i.e., orthogonality of is preserved for t [, ]. Similar arguments can be used to show Y remains orthonormal for t [, ]. Next, we show that the matrix W in ODE system 7 is anti-symmetric. Consider two pairs of indices i, j and p, q with i > j and p > q. he corresponding equations for χ i,j and χ p,q are dχ i,j dχ p,q m m = D ik χ k,j D jl χ i,l, = k=,k i,j m l=,l i,j m D pk χ k,q k=,k p,q l=,l p,q D ql χ p,l, 2a 2b where we have identified χ i,j with χ j,i. First, we consider the case two index pairs are identical. We see from Eq. 2a that there is no χ i,j on the right side, so W i,j,i,j =. Next, we consider the case none of indices in one index pair is equal to any in the other, i.e., i p, q and j p, q. It is obvious that no χ p,q appears in the summations on the right side of Eq. 2a, which implies W i,j,p,q =. Similarly, we have W p,q,i,j =. Last, we consider the case in which one index in one index pair is equal to one index in another index pair but the remaining two indices from two index pair are not equal. Without loss of generality, we assume i = p and j q. From Eq. 2a, we have dχ i,j = D jq χ i,q = D jq χ p,q, where we have intentionally isolated the relevant term from the second summation and the last equality is due to i = p. Similarly, from Eq. 2b, we have dχ p,q = D qj χ p,j = D qj χ i,j. he above two equations imply that W i,j,p,q = W p,q,i,j. Similar arguments apply to other cases. his completes the proof Error Analysis In this section, we will consider the convergence of the solution obtained by the DyBO formulation 6 to the original SPDE. heorem 3.2 gives the error analysis of the DyBO formulation. 9

10 heorem 3.2 he consistency between the DyBO formulation and the original SPDE. he stochastic solution of system 6 satisfies the following modified SPDE ũ = Lũ + e m, 2 where e m is the error due to the m-term truncation and e m = Π Y Π Lũ. 22 Proof. We show consistency by computing directly from Eqs. 6band 6c or equivalently Eqs.?? and??, [ ] Lũ Λ, Lũ Λ, E LũY Y, = Lũ + D Y Y = CY + YE Combining the above two, we have where e m = = Lũ Λ, Lũ [ LũY ] Y E [ LũY ], Lũ Λ, Lũ + E + Y = Lũ + e m, Λ. + D + C Y Λ [ ], E LũY Y [ ] [ ] + YE LũY Y E LũY, ], Lũ Y Y, [ Lũ Λ where Eq. 4c has been used in deriving the first equality. Eq. 2. Λ By noticing Lũ = Lũ E [Lũ], we prove Remark 3.2. According to heorem 3., the bi-orthogonality of and Y are preserved for all time. Because both Hilbert space L 2 D and L 2 Ω are separable, the spatial basis and the stochastic basis Y become a complete basis for L 2 D and L 2 Ω as m +, respectively, which implies lim m + e m =. Next we consider a special case where the differential operator L is deterministic and linear, such as Lv = cx v, or Lv = i a ij x v j, where cx and a ij x are deterministic functions. Only initial conditions are assumed to be random, i.e., the randomness propagates into the system only through initial conditions. Corollary 3.3. Let the differential operator L be deterministic and linear. he residual in Eq. 2 is zero for all time, i.e., e m t, t >. Proof. his can be seen by directly computing the truncation error e m. First we substitute Lũ = Lũ E [Lũ] into Eq. 22, e m = Π Y Lũ E [Lũ] Λ, Lũ E [Lũ]. Since the differential operator L is deterministic and linear, we have Lũ = Lū + LY, where L = Lu, Lu 2,, Lu m. his gives e m = Π Y Lū + LY E [ Lū + LY ] Λ, Lũ E [Lũ] = Π Y LY Λ, L Y = Π Y Y L L, Λ =. he last line is due to the orthogonal complementary project.

11 he above corollary implies that DyBO is exact if the randomness can be expressed in a finite-term KL expansion. he next corollary concerns a slightly different case where the differential operator L is affine in the sense that Lu = Lu+fx, t, ω and the differential operator L is linear and deterministic. he stochastic force f, t, L 2 D Ω for all t. Corollary 3.4. If the differential operator L is affine, i.e., Lu = Lu + fx, t, ω, and f is a second-order stochastic process at each fixed time t, the residual in Eq. 22 is given below e m = Π Y Π f. Proof. Again by directly computing, we have by the linearity of the differential operator L Simple calculation shows that Substituting into eqn. 22, we complete the proof. Lũ = Lū + LY + f. Lũ = Lũ E [Lũ] = LY + f E [f]. Remark 3.3. For this special case, Corollary 3.4 implies that numerical solutions are accurate as long as the spatial basis and the stochastic basis Y provide good approximations to the external forcing term f, which is not surprising at all. 4. Numerical Implementation 4.. Representation of Stochastic Basis Y he DyBO formulation 6 is a combination of deterministic PDEs 6a and 6b and SODEs 6c. For the deterministic PDEs in the DyBO formulation, we can apply suitable spatial discretization schemes and numerical integrators to solve them numerically. One of the conceptual difficulties, from the viewpoint of the classical numerical PDEs, involves representations of random variables or functions defined on abstract probability space Ω. Essentially, there are three ways to represent numerically stochastic basis Yω, t: ensemble representations in sampling methods, such as Monte carlo method, sparse grid based stochastic collocation method [, 8] and spectral representations, such as the generalized Polynomial Chaos method [54, 55, 5, 28, 34] or Wavelet based Chaos method [36] Ensemble Representation DyBO-MC In the MC method framework, the stochastic basis Yω, t are represented by an ensemble of realizations, i.e., Yω, t {Yω, t, Yω 2, t, Yω 3, t,, Yω Nr, t}, where N r is the total number of realizations. hen the expectations in the DyBO formulation 6 can be replaced by ensemble averages, i.e., x, t = N r N r i= Lũx, t, ω i, x, t = x, tdt + N r ω i, t = Yω i, tct + N r i= Lũx, t, ω i Yω i, t, Lũx, t, ωi, x, t Λ t, 23a 23b 23c

12 where Ct and Dt can be solved from 5 with G ū,, Y = Λ, N r N r i= Lũx, t, ω i Yω i, t, 24 and Lũx, t, ω i = Lũx, t, ω i N r N r i= Lũx, t, ω i. 25 he above system is the MC version of DyBO method and we denote it as DyBO-MC. Close examinations reveal that Eq. 23a for expectation and Eq. 23b for spatial basis are only solved once for all the realizations at each time iteration while Eq. 23c for the stochastic basis is decoupled from realization to realization and can be solved simultaneously across all realizations. In this sense, we see that DyBO-MC is semisampling and it preserves some desirable features of MC method, for example, the convergence rate does not depend on stochastic dimensionality and the solution process of Y ω i, t s are decoupled. Clearly, not only the number of realizations, but also how these realizations {ω i } Nr i= are chosen in Ω affects the numerical accuracy and the efficiency. Quasi Monte Carlo method [43, 4], variance reduction and other techniques may be combined into DyBO-MC to further improve its efficiency and accuracy. However, the disadvantage of MC method is its slow convergence. Developing effective sampling version of DyBO method such as the multi-level Monte Carlo method [24] is currently under investigation Spectral Representation DyBO-gPC In many physical and engineering problems, randomness generally comes from various independent sources, so randomness in SPDE is often given in terms of independent random variables ξ i ω. hroughout this paper, we assume only a finite number of independent random variables are involved, i.e., ξω = ξ ω, ξ 2 ω,, ξ r ω, where r is the number of such random variables. Without loss of generality, we can further assume they all have identical distribution ρ. hus, the solution of SPDE is a functional of these random variables, i.e., ux, t, ω = ux, t, ξω. When SPDE is driven by some known stochastic process, such as Brownian Motion B t, by Cameron-Martin theorem [9], the stochastic solution can be approximated by a functional of identical independent standard Gaussian variables, i.e., ux, t, ω ux, t, ξω with ξ i being a standard normal random variable. In this paper, we only consider the case where randomness is given in terms of independent standard Gaussian random variables, i.e., ξ i N, and H are a set of tensor products of Hermite polynomials. In this case, the method described above is also known as the Wiener-Chaos Expansion [52, 9, 2, 3, 28]. When the distribution of ξ i is not Gaussian, the discussions remain almost identical and the major difference is the set of orthogonal polynomials. We denote by {H i ξ} i= the one-dimensional polynomials orthogonal to each other with respect to the common distribution ρz, i.e., H i ξh j ξρξ dξ = δ ij. For some common distributions, such as Gaussian or uniform distribution, such polynomial sets are wellknown and well-studied, many of which fall in the Ashley scheme, see [54]. For general distributions, such polynomial sets can be obtained by numerical methods, see [5]. Furthermore, by a tensor product representation, we can use the one-dimensional polynomial H i ξ to construct a complete orthonormal basis H α ξ s of L 2 Ω as follows r H α ξ = H αi ξ i, α J r, where α is a multi-index, i.e., a row vector of non-negative integers, i= α = α, α 2,, α r α i N, α i, i =, 2,, r, 2

13 and J r is a multi-index set of countable cardinality, J r = {α = α, α 2,, α r α i, α i N} \ {}. We have intentionally removed the zero multi-index corresponding to H ξ = since it is associated with the mean of the stochastic solution and it is better to deal with it separately according to our experience. When no ambiguity arises, we simply write the multi-index set J r as J. Clearly, the cardinality of J r is infinite. For the purpose of numerical computations, we prefer a polynomial set of finite size. here are many choices of truncations, such as the set of polynomials whose total orders are at most p, i.e., { } r J p r = α α = α, α 2,, α r, α i, α i N, α = α i p \ {}, and sparse truncations proposed in [28], see also Luo s thesis [33]. Again, we may simply write such a truncated set as J when no ambiguity arises. he cardinality of J, or the total number of polynomial basis functions, denoted by N p = J, is equal to p+r! p!r!. As we can see, N p grows exponentially fast as p and r increase. his is also known as the curse of dimensionality. By the Cameron-Martin theorem [9], we know that the solution of SPDE admits a generalized Polynomial Chaos expansion gpce ux, t, ω = vx, t + v α x, th α ξω vx, t + v α x, th α ξω. 26 α J α J i= If we write H J ξ = H α ξ, H α2 ξ,, H αnp ξ Vx, t = v α x, t, v α2 x, t,, v αnp x, t α i J, α i J, both of which are row vectors, the above gpce can be compactly written in a vector form u gpc x, t, ω = vx, t, ω = vx, t + Vx, thξ. 27 By substituting the above gpce into Eq. a, it is easy to derive the gpc formulation for SPDE, v = E [Lv], 28a V [ ] = E LvH. 28b Now, we turn our attention to the gpc version of DyBO formulation DyBO-gPC. he Cameron-Martin theorem also implies the stochastic basis Y i ω, t s in the KL expansion 7 can be approximated by the linear combination of polynomials chaos, i.e., Y i ω, t = α J H α ξωa αi t, i =, 2,, m, 29 or in a matrix form, Yω, t = H ξω A, 3 where A R Np m. he expansion 7 now reads ũ = ū + A H. 3

14 We can derive equations for ū, and A, instead of ū, and Y. In other words, the stochastic basis Y are identified with a matrix A R Np m in DyBO-gPC. Substituting Eq. 3 into Eq. 6c and using the identity E [ H H ] = I, we get the DyBO-gPC formulation of SPDE, = E [Lũ], 3a [ ] = D + E LũH A, 3b da ] = AC + E [H Lũ, Λ, 3c where Ct and Dt can be solved from 5 with [ ] [ ] G ū,, Y = Λ, E LũY = Λ, E LũH A. 32 By solving the system 3, we have an approximate solution to SPDE or simply u DyBO when no ambiguity arises. u DyBO-gPC = ū + A H, gsc Representation DyBO-gSC In our DyBO-gPC method, the stochastic process Y i ξω, t is projected on to the gpc basis H and replaced by gpc coefficients, i.e., A αi, which can be computed by A αi = E [Y i ωh α ξω]. Numerically, the above integral can be evaluated with high accuracy on some sparse grid. In other words, the stochastic basis Y can also be represented as an ensemble of realizations Yω i where ξω i s are nodes from some sparse grid and associated with certain weight w i. he gsc version of DyBO formulation, which we denote as DyBO-gSC, is Ns x, t = w i Lũx, t, ω i, i= x, t = x, N s tdt + w i Lũx, t, ωi Yω i, t, ω i, t = Yω i, tct + i= Lũx, t, ωi, x, t Λ t, i =, 2,..., N s, where {ω i } Ns i= are the sparse grid points, {w i} Ns i= are the weight, N s is the number of sparse grid points and Ct, Dt can be solved from 5 with G ū,, Y = Λ N s, w i Lũx, t, ωi Yω i, t. 34 i= he primary focus of this paper is on DyBO-gPC methods. We will provide details and numerical examples of DyBO-gSC methods in our future work Eigenvalue Crossings As the system evolves, eigenvalues of different basis in the KL expansion of the SPDE solution may increase or decrease. Some of them may approach each other at some time, cross and then separate as 4

15 -Stage x, t x, s Eigenvalue λi λ τ < δτ out λ2 Y-Stage Yω, t Yω, s Entering: τ < δ τ in Exiting: τ > δ τ out s λ3 s s + ky s + 3kY t t 2 ime t Figure 2: Illustration of Eigenvalue Crossing illustrated in Fig. 2. In the figure, λ and λ 2 cross each other at t and λ and λ 3 cross each other at t 2. In this case, if C and D continue to be solved from the linear system 4 via 5, numerical errors will pollute the results. Here, we propose to freeze or Y temporarily for a short time and continue to evolve the system using different equations as derived below. At the end of this short duration, the solution is recast into the bi-orthogonal form via the KL expansion, which can be achieved efficiently since or Y is still kept orthogonal in this short duration. We are also currently exploring alternative approach to overcome the eigenvalue crossing problem, and will report our result in a subsequent paper. In order to apply the above strategy, we have to be able to detect that two eigenvalues may potentially cross each other in the near future. here are several ways to detect such crossing. Here we propose to monitor the following quantity τ = min i j λ i λ j maxλ i, λ j. 35 Once this quantity drops below certain threshold δτ in,, we adopt the algorithm of freezing or Y to evolve the system and continue to monitor this quantity τ. When τ exceeds some pre-specified threshold δτ out,, the algorithm recasts the solution in the bi-orthogonal form via an efficient algorithm detailed in the next two sub-sections and continues to evolve the DyBO system. hreshold δτ in and δτ out may be problem-dependent. In our numerical experiments, we found δτ in = % and δτ out = % gave accurate results. Here we only describe the basic idea of freezing stochastic basis Y or Y-Stage algorithm. he method of freezing the spatial basis or -Stage algorithm can be obtained analogously, see [] for more details. Suppose at some time t = s, potential eigenvalue crossing is detected and the stochastic basis Y are frozen for a short duration s, i.e., Yω, t Yω, s for t [s, s + s], which we call Y-stage. Because Y are orthonormal, the solution of SPDE admits the following approximation of a truncated expansion. ũx, t, ω = ūx, t + m u i x, ty i ω, s = ūx, t + x, tyω, s. 36 i= It is easy to derive a new evolution system for this stage, = E [Lũ], 37a [ ] = E LũY. 37b During this stage, Y is unchanged and orthogonal, but changes in time and would not maintain its orthogonality. he solution is no longer bi-orthogonal, so eigenvalues cannot be computed via λ i = u i 2 L 2 D 5

16 any more. Next, we derive formula for eigenvalues in this stage and then show how the solutions are recast into the bi-orthogonal form at the end of the stage, i.e., t = s + s. he covariance function can be computed as Covũx, y = E [ũx ūx ũy ūy] = E [ xy Y y ] = x y, where t is omitted for simplicity and E [ Y Y ] = I is used. By some stable orthogonalization procedures, such as the modified Gram-Schmi algorithm, x can be written as x = QxR, 38 where Qx = q x, q 2 x,, q m x, q i x L 2 D for i =, 2,, m, Q x, Qx = I and R R m m. Here RR R m m is a positive definite symmetric matrix and its SVD decomposition reads RR = WΛ R W, 39 where W R m m is an orthonormal matrix, i.e., WW = W W = I, and Λ R is a diagonal matrix with positive diagonal entries. he computational cost of W and Λ R is negligible compared to other parts of the algorithm. he covariance function can be rewritten as Covũx, y = QxRR Q y = QxWΛ R W Q y. Now, it is easy to see that the eigenfunctions of covariance function Covũx, y are Ũx = QxWΛ 2 R, 4 and eigenvalues are diag Λ R because Ũ Ũ, = Λ 2R W Q, QWΛ 2R = Λ R, where we have used the orthogonality of Q and W. o compute Ỹ, we start with the identity ỸŨ = Y, and multiply row vector Ũ on both sides from the right and take inner product,, Ũ Ỹ, Ũ = Y, Ũ, where Ũ Ũ, = Λ R,, Ũ = R Q x, QxWΛ 2R = R WΛ 2 R. So we obtain the stochastic basis Ỹ, which is given by Ỹ = YR WΛ 2 R. 4 If a generalized polynomial basis H is adopted to represent Y, we freeze A instead, i.e., At As and the system 37 is replaced by = E [Lũ], 42a [ ] = E LũH A, 42b where At As. At the exiting point, Eq. 4 is replaced by à = AR WΛ 2 R. 43 As we can see, the computation of eigenvalues in this stage is not trivial, so we do not want to compute τ every time iteration. Instead, τ is only evaluated every k Y time steps to achieve a balance between computational efficiency and accuracy. See the zoom-in figure in Fig. 2 for illustration. 6

17 5. Numerical Examples Previous sections highlight the analytical aspects of the DyBO formulation and its numerical algorithm. In this section, we demonstrate the effectiveness of this method by several numerical examples with increasing level of difficulties, each of which emphasizes and verifies some of analytical results in the previous sections. In the first example, we consider a SPDE which is driven purely by stochastic forces. he purpose of this example is to show that the DyBO formulation tracks the KL expansion. Corollary 3.3 regarding the error propagation of the DyBO method is numerically verified in the second numerical example where a transport equation with a deterministic velocity and random initial conditions is considered. In the last example, we consider Burgers equation driven by a stochastic force as an example of a nonlinear PDE driven by stochastic forces. We demonstrate the convergence of the DyBO method with respect to the number of basis pairs, m. In the second part of this paper [], we will consider the more challenging Navier-Stokes equations and the Boussinesq approximation with random forcings. Some important numerical issues, such as adaptivity, parallelization and computational complexity, will be also studied in details. 5.. SPDE Purely Driven by Stochastic Force o study the convergence of our DyBO-gPC algorithms, it is desirable to construct a SPDE whose solution and KL expansion are known analytically. his would allow us to perform a thorough convergence study for our algorithm especially when we encounter eigenvalue crossings. In this example, we consider u a SPDE which is purely driven by a stochastic force f, i.e., = Lu = fx, t, ω. he solution can be obtained by direct integration, i.e., ux, t, ω = ux,, ω + t fx, s, ω ds. Specifically, in this section, we consider the SPDE u = Lu = fx, t, ξω, x D = [, ], t [, ], 44 where ξ = ξ, ξ 2,, ξ r are independent standard Gaussian random variables, i.e., ξ i N,, i =, 2,, r. he exact solution is constructed as follows ux, t, ξ = vx, t + Vx, tz ξ, t, 45 where Vx, t = VxW V tλ 2 V t, Zξ, t = ZξW Z t, Vx = v x,, v m x with v i x, v j x = δ ij and Zξ = Z ξ,, Z ] m ξ with E [ Zi Zj = δ ij for i, j =, 2,, m. W V t and W Z t are m-by-m orthonormal matrices, and Λ 2 V t is a diagonal matrix. Clearly, the exact solution u is intentionally given in the form of its finite-term KL expansion and the diagonal entries of matrix Λ V t are its eigenvalues. By carefully choosing the eigenvalues, we can mimic various difficulties which may arise in more involved situations and devise corresponding strategies. he stochastic forcing term on the right hand side of Eq. 44 can be obtained by differentiating the exact solution, i.e., Lu = f = v + V Z + V dz. 46 he initial condition of SPDE 44 can be obtained by simply setting t = in Eq. 45. From the general DyBO-gPC formulation 6, simple calculations give the DyBO-gPC formulation for SPDE 44 = v, V = D + W Z + V dw Z da [ ] = AC + E H V Z W Z 47a [ Z ] E H A, 47b + dw Z V, Λ, 47c 7

18 and he initial conditions are simply G u,, A = Λ, V W Z + V dw [ Z ] Z E H A. ūx, = vx,, x, = Vx,, A = E [ H Zξ, ] = E In the event of eigenvalue crossings, we have the Y-stage system from 42, or the -stage system, = v, = V W Z + V dw Z = v, da [ ] = E H V Z W Z [ ] H Z W Z. 48a [ Z ] E H A, 48b + dw Z V, Λ. In the numerical examples presented in this section, we consider a small system m = 3 and use the following settings, Vx = 2 sinπx, 2 sin5πx, 2 sin9πx, Zx = H ξ, H 2 ξ, H 3 ξ, cos b V t sin b V t cos b Z t sin b Z t W V t = P V sin b V t cos b V t P V, W Z t = P Z sin b Z t cos b Z t P Z, where b V = 2., b Z = 2., P V and P Z are two orthonormal matrices generated randomly, and H ξ = H ξ, H 2 ξ,, H 5 ξ. Eigenvalues in the KL expansion of an SPDE solution may increase or decrease as time increases. Some of them may approach each other at some time, cross and then separate later. When two eigenvalues are close or equal to each other, numerical instability may arise in solving matrices C and D via 5. In Section 4.2, we have proposed a -Stage by freezing the spatial basis or a Y-Stage by freezing the stochastic basis Y temporarily to resolve this issue. Here, we demonstrate the success by incorporating such strategies into the DyBO algorithm. o this end, we choose =.2 and eigenvalues Λ 2 V = diag sin 2πt + 2, cos 2πt +.5,.8, where eigenvalues cross each other at t.675,.25,.532,.7985,.965,.675, see Fig. 5. In this numerical example, the time step t =. 3 and k = 2 in the -stage, i.e., the exiting condition is checked every 2 time iterations when the system is in the -stage. he mean E [u], the standard deviation SD Var u and the three spatial basis in KLE at time t = are shown in Fig. 3 and Fig. 4, respectively. Clearly, the results given by DyBO almost perfectly match the exact ones with L 2 relative errors of mean and SD below since only the spatial and temporal discretizations contribute to the numerical errors in this example. In Fig. 5, eigenvalues are plotted as functions of time and zoom-in figures are given at t =.675 and t =.7985 to show eigenvalue crossings and invoking of the -stage algorithm. We also check the deviation of the computed spatial and stochastic basis from the bi-orthogonality by monitoring u i, u j u i L 2 D u j L 2 D 49a 49b for the spatial basis and E [Y i Y j ] for the stochastic basis. he results are shown in Fig. 6. We observe that the deviation of the spatial basis from orthogonality is very small < throughout the computation. Large deviations of the stochastic basis from the orthogonality only occur when eigenvalues cross each other since the DyBO algorithm is not designed to preserve the orthogonality of Y during the -stage and orthogonality is only restored once the algorithm exits from the -stage. We have also applied the Y-stage to overcome eigenvalue-crossing issues in this numerical examples and the results are very similar. 8

19 Mean Mean Exact Exact.4.4DyBO DyBO SD SD Exact Exact DyBO DyBO Figure 3: Mean and SD computed by DyBO at time t = Exact DyBO Exact DyBO Exact DyBO Exact DyBO Exact DyBO Exact DyBO Exact DyBO4 Exact DyBO Exact DyBO Figure 4: he spatial basis u x, t, u 2 x, t, u 3 x, t from the left to the right in KL expansion of SPDE solution computed by DyBO at time t =.2. Stage Stage λ λ 2 λ x u & u 2 u 2 & u 3 u 3 & u Figure 5: Eigenvalues computed by DyBO. wo zoom-in 2 figures are provided when eigenvalues cross λ i x EY.5 2 u & u 2 u & u 2 3 u & u λ i Y & Y 2. 8 Y 2 & Y Y & Y EY a Orthogonality of the spatial basis x, t..5 x ubar L 2 Error b Orthogonality of the L 2 stochastic Error basis Yω, t Figure 6: Bi-orthgonality of spatial and.5 stochastic basis Y & Y 2 Y 2 & Y 3 Y & Y 3.5 x ubar L 2 Error.8 L 2 Error.5.6.4

20 5.2. Linear Deterministic Differential Operators with Random Initial Conditions In this section, we consider a linear PDE with random initial conditions, i.e., the differential operator L is deterministic and linear. For simplicity, we assume periodic boundary conditions. u = Lu, x [, ], t [,.4], 5a u, t, ξ = u, t, ξ, 5b ux,, ξ = ůx, ξ, where ξ = ξ, ξ 2,, ξ r are independent standard Gaussian random variables. Consider the gpc expansion of solution u = v + VH. From Eq. 28, it is easy to obtain the gpc formulation of this SPDE, v = L v, 5a V = LV, 5b where the initial conditions vx, = E [ů] and Vx, = E [ůxh]. Clearly, the gpc formulation 5 is linear and provides the exact solution to the original SPDE 5 if the finite-term gpc expansion of the initial condition is exact, i.e., ů = E [ů] + E [ůh] H. Now consider the KL expansion of the solution, u = ū + Y = ū + A H. From the general DyBO-gPC formulation 6, simple calculations give the DyBO-gPC formulation for the SPDE 5 = Lū, 52a = D + L, 52b da = AC + A L, Λ, 52c where C and D can be computed via Eq. 5 from G = Λ, L. If we compare the DyBO formulation 52 and the gpc formulation 5 for the linear SPDE 5, we observe the following interesting phenomenon. Although the original SPDE is linear and the gpc formulation remains linear, the DyBO formulation is clearly not linear. However, this is not surprising because KL expansion is not a linear procedure. On the other hand, the gpc expansion is indeed a linear procedure. As we argue in Sec. 3.2, our DyBO formulation essentially tracks the KL expansion of the exact solution. herefore, the non-linearity of KLE must be naturally built into the DyBO formulation to enable such tracking. Corollary 3.3 implies that the solution given by the DyBO formulation is exact if the initial condition ů can be expressed exactly by a finite-term KL expansion. o verify this numerically, we consider a transport equation, i.e., L = sin2πx + 2t and the initial condition is a functional of three standard Gaussian random variables, i.e., r = 3, ůx, ξ = cos2πx + sin2πx, 2 sin4πx, 3 sin6πx Å H J, where J = { α α J 4 3, α 3 3 } \ {}. Matrix Å is an orthonormal matrix generated randomly at the beginning of our simulation. With time step t =. 3 and spatial grid size x = /28, the elements in the spatial basis computed by DyBO match perfectly the exact ones given by gpc as shown in Fig. 7. In Fig. 8, the L 2 relative error of SD is plotted as a function of time t and remains very small 8. o confirm such errors are introduced mainly by the spatial and the temporal discretizations, we use another two sets of finer grid sizes, t =.5 3, x = /256, and t =.25 3, x = /52, and repeat the computations. he SD error drops significantly since we use spectral methods and a fourth order RK method, which essentially verifies numerically Corollary c