Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University
Discretization of Boundary Conditions Discretization of Boundary Conditions On boundary nodes and irregular interior nodes, we usually need to construct different finite difference approximation schemes to cope with the boundary conditions. Remember that the set of irregular interior nodes is given by J Ω = {j J \ J D : D Lh (j) J}, that is J Ω is the set of all such interior node which has at least one neighboring node not located in Ω. For simplicity, we take the standard 5-point difference scheme for the 2-D Poisson equation u = f as an example to see how the boundary conditions are handled. 2 / 36
Discretization of the Dirichlet Boundary Condition Since N, E are not in J, P is a irregular interior node, on which we need to construct a difference equation using the Dirichlet boundary condition on the nearby points N, P and/or E. The simplest way is to apply interpolations. N N * n P * W w P e E * E Ω D s S
Discretization of the Dirichlet Boundary Condition Difference equations on P derived by interpolations: Zero order: U P = U P with truncation error O(h); First order: U P = hx U E +h x U W h x +hx truncation error O(h 2 ); or U P = hy U N +h y U S h y +h, with y N N * n P * W w P e E * E Ω D s S
Discretization of the Dirichlet Boundary Condition Difference equations on P can be derived by extrapolations and the standard 5-point difference scheme: The grid function values on the ghost nodes N and E can be given by second order extrapolations using the grid function values on S, P, N and W, P, E respectively (see Exercise 1.3). N N * n P * W w P e E * E Ω D s S
Discretization of the Dirichlet Boundary Condition Difference equations on P can also be derived by the Taylor series expansions and the partial differential equation to be solved: Express u W, u E, u S, u N by the Taylor expansions of u at P. Express u x, u y, u xx, u yy on P in terms of u W, u E, u S, u N and u P. Substitute these approximation values into the differential equation (see Exercise 1.4). N N * n P * W w P e E * E Ω D s S
Discretization of the Dirichlet Boundary Condition Finite difference schemes with nonuniform grid spacing: a difference equation on P using the values of U on the nodes N, S, W, E and P with truncation error O(h): { ( 2 UE U P h x +hx hx U ) P U W + 2 ( UN U P h x h y +hy hy U )} P U S = f P. h y N Shortcoming: nonsymmetric. n N * P * W w P e E * E Ω D s S
Discretization of the Dirichlet Boundary Condition Symmetric finite difference schemes with nonuniform grid spacing: a difference equation on P using the values of U on the nodes N, S, W, E and P with truncation error O(1): { ( 1 UE U P h x h x U P U W h x ) + 1 ( UN U P h y hy U P U S h y )} = f P. N It can be shown: the global error is O(h 2 ). n N * P * W w P e E * E Ω D s S
Discretization of the Dirichlet Boundary Condition Construct a finite difference equation on P based on the integral form of the partial differential equation u V P ν ds = V P f dx: ( UW U P + U E U ) P hy + φhy h x hx 2 ( US U P + U N U ) P hx + θhx (h x + θhx)(h y + φhy ) h y hy = f P, 2 4 where θhx/2, φhy /2 are the lengthes of the line segments Pe and Pn. (O(h), nonsymmetric) W w N n P N * e P * E * E Ω D s S
Discretization of Boundary Conditions Discretization of the Dirichlet Boundary Condition Extension of the Dirichlet Boundary Condition Nodes J D Add all of the Dirichlet boundary points used in the equations on the irregular interior nodes concerning the curved Dirichlet boundary, such as E, N and P, into the set J D to form an extended set of Dirichlet boundary nodes, still denoted by J D. N N * n P * W w P e E * E Ω D s S 10 / 36
Discretization of the Neumman Boundary Condition Since N, E are not in J, P is a irregular interior node, on which we need to construct a difference equation using the Nuemman boundary condition on the nearby points N, P and/or E. The simplest way is again to apply interpolations. N N w a N * n w P * W ξ w P E * E s Ω N s w η b S w S
Discretization of the Neumman Boundary Condition Let P be the closest point to P on Ω N, and α be the angle between the x-axis and the out normal to Ω N at the point P. ν u(p ) u(p) ν P, a zero order extrapolation to the out normal, leads to a difference equation on P with local truncation error O(h) : U P U W h x cos α + U P U S h y sin α = g(p ). N N w a N * n w P * W ξ w P E * E s Ω N s w η b S w S
Discretization of the Neumman Boundary Condition We can combine the nonuniform grid spacing difference equations { ( 2 UE U P U ) P U W + 2 ( UN U P U )} P U S = f h x +hx hx h x h y +hy hy P, h y or { ( 1 UE U P U ) P U W + 1 ( UN U P U )} P U S = f h x hx h x h y hy P, h y on the irregular interior node P, and add in the difference equations for the new unknowns U N and U E by making use of the boundary conditions. Say N w a N N * U N U ξ ξn = g(n ), O(h), n w P * and U E U η ηe = g(e ), O(h). W ξ s w w P s η E * Ω N b E S w S
Discretization of the Neumman Boundary Condition The finite volume method based on the integral form of the Poisson equation u V P ν ds = V P f dx with V P being the domain enclosed by the broken line segments, where an w PN W, leads to an asymmetric finite volume scheme on the irregular interior node P U N W U P an w U W U P h y U S U P s w b g(p N W P h x h ) ãb = f (P) V P. y N The local truncation error is O(h), since numerical quadrature is not centered. N w W ξ n w w a P N * P * E * E s w s η Ω N b S w S
Discretization of Boundary Conditions Discretization of the Neumman Boundary Condition More Emphasis on Global Properties In dealing with the boundary conditions, compared with the local truncation error, more attention should be put on the more important global features: Symmetry; Maximum principle; Conservation; etc.. 15 / 36
Discretization of Boundary Conditions Discretization of the Neumman Boundary Condition More Emphasis on Global Properties so that the finite difference approximation solution can have better stability and higher order of global convergence; inherit as much as possible the important global properties from the analytical solution; be solved by applying fast solvers. 16 / 36
Truncation Error, Consistency, Stability and Convergence Truncation Error, Consistency, Stability and Convergence Consider the boundary value problem of a partial differential equation { Lu(x) = f (x), x Ω, Gu(x) = g(x), x Ω and the corresponding finite difference approximation equation defined on a rectangular grid with spacing h L h U j = f j, j J. Notice, if j is not a regular interior node, then L h and f j may depend on G and g as well as on L and f. Denote Lu(x) = Lu(x), if x Ω, and Lu(x) = Gu(x), if x Ω. 17 / 36
Truncation Error, Consistency, Stability and Convergence Truncation Error and consistency Truncation Error Definition Suppose that the solution u to the problem is sufficiently smooth. Let T j (u) = L h u j ( Lu) j, j J. Define T j (u) as the local truncation error of the finite difference operator L h approximating to the differential operator L. The grid function T h (u) = {T j (u)} j J is called the truncation error of the finite difference equation approximating to the problem. Remark 1: Briefly speaking, the truncation error measures the difference between the difference operator and the differential operator on smooth functions. Remark 2: T h (u) can also be viewed as a piece-wise constant function defined on Ω via the control volumes. 18 / 36
Truncation Error, Consistency, Stability and Convergence Truncation Error and consistency Point-Wise Consistency of L h Definition The difference operator L h is said to be consistent with the differential operator L on Ω, if for all sufficiently smooth solutions u, we have lim T j(u) = 0, j J Ω. h 0 The difference operator L h is said to be consistent with the differential operator L on the boundary Ω, if for all sufficiently smooth u, we have lim T j(u) = 0, j J \ J Ω. (1) h 0 Remark: Briefly speaking, this is the point-wise consistency of L h to L. In fact, for u sufficiently smooth, in the above definition lim h 0 T h (u) = 0 in either l or L are equivalent. 19 / 36
Truncation Error, Consistency, Stability and Convergence Truncation Error and consistency Consistency and accuracy in the norm Definition The finite difference equation L h U = f h is said to be consistent in the norm with the boundary value problem of the differential equation Lu = f, if, for all sufficiently smooth u, we have lim T h(u) = 0. h 0 The truncation error is said to be of order p, or order p accurate, if the convergent rate above is of O(h p ), i.e. T h (u) = O(h p ). Remark: Here T h (u) is viewed as a piece- wise constant function defined on Ω via the control volumes. The norms in the above definition is the corresponding function norm. 20 / 36
Truncation Error, Consistency, Stability and Convergence Stability Stability in the norm Definition The difference equation L h U = f is said to be stable or have stability in the norm, if there exists a constant K independent of the grid size h such that, for arbitrary grid functions f 1 and f 2, the corresponding solutions U 1 and U 2 to the equation satisfy U 1 U 2 K f 1 f 2, h > 0. The stability implies the uniform well-posedness of the difference equation, more precisely, it has a unique solution which depends uniformly (with respect to h) Lipschitz continuously on the right hand side (source terms and boundary conditions). 21 / 36
Truncation Error, Consistency, Stability and Convergence Convergence Convergence in the norm Definition The difference equation L h U = f is said to converge in the norm to the boundary value problem Lu = f, or convergent, if, for any given f (f and g) so that the problem Lu = f is well posed, the error e h = {e j } j J {U j u(x j )} j J of the finite difference approximation solution U satisfies lim e h = 0. h 0 Furthermore, if e h = O(h p ), then the difference equation is said to converge in order p, or order p convergent. 22 / 36
Truncation Error, Consistency, Stability and Convergence Convergence Stability + Consistency Convergence Since L h e j = (L h U j L h u j ), the stability of the difference operator L h yields U u = e h K L h U L h u K( L h U Lu + Lu L h u ). 1 L h U Lu is the residual of the algebraic equation L h U = f, which is 0 when U is the solution of the difference equation; 2 L h u Lu = T h is the truncation error; 3 If U is the finite difference solution, then, the stability implies U u = e h K T h. 4 Stability + Consistency Convergence. 23 / 36
Truncation Error, Consistency, Stability and Convergence Convergence The Convergence Theorem Theorem Suppose that the finite difference approximation equation L h U = f of the boundary value problem of partial differential equation Lu = f is consistent and stable. Suppose the solution u of the problem Lu = f is sufficiently smooth. Then the corresponding finite difference equation must converge, and the convergent order is at least the order of the truncation error, i.e. T h = O(h p ) implies e h = O(h p ). Note the additional condition that the solution u is sufficiently smooth, which guarantees that the truncation error for this specified function converges to zero in the expected rate. 24 / 36
Error Analysis Based on the Maximum Principle The Maximum Principle The Problem and Notations for the Maximum Principle 1 Ω R n : a connected region; 2 J = J Ω J D : grid nodes with grid spacing h; 3 Boundary value problem of linear difference equations: { L h U j = f j, j J Ω, U j = g j, j J D, 4 L h has the following form on J Ω : L h U j = c ij U i c j U j, j J Ω. i J\{j} 5 D Lh (j) = {i J \ {j} : c ij 0}: the set of neighboring nodes of j with respect to L h. 25 / 36
Error Analysis Based on the Maximum Principle The Maximum Principle Connection and J D connection of J with respect to L h Definition A grid J is said to be connected with respect to the difference operator L h, if for any given nodes j J Ω and i J, there exists a set of interior nodes {j k } m k=1 J Ω such that j 0 = j, i D Lh (j m ), j k+1 D Lh (j k ), k = 0, 1,..., m 1. Suppose J D, a grid J is said to be J D connected with respect to the difference operator L h, if for any given interior node j J Ω there exists a Dirichlet boundary node i J D and a set of interior nodes {j k } m k=0 J Ω such that the above inclusion relations hold. 26 / 36
The Maximum Principle Theorem Suppose L h U j = i J\{j} c iju i c j U j, j J Ω ; J and L h satisfy (1) J D, and J is J D connected with respect to L h ; (2) c j > 0, c ij > 0, i D Lh (j), and c j c ij. i D Lh (j) Suppose the grid function U satisfies L h U j 0, j J Ω. Then, { } M Ω max U i max max U i, 0. i J Ω i J D Furthermore, if J and L h satisfy (3): J is connected with respect to L h ; and there exists interior node j J Ω such that U j = max i J U i 0. Then, U must be a constant on J.
Error Analysis Based on the Maximum Principle The Maximum Principle Proof of The Maximum Principle Assume for some j J Ω, U j = M Ω > M D max i JD U i, M Ω > 0. By the J D connection, there exist i J D and {j k } m k=0 J Ω such that the inclusion relation j k+1 D Lh (j k ) hold. It follows from the conditions on L h and the condition (2) that U j c ij max c Uˆl c ij M Ω. j ˆl D Lh (j) c j i D Lh (j) c ij c j i D Lh (j) 1, this implies that the Since U j = M Ω 0 and i D Lh (j) equalities must all hold, which can be true only if L h U j = 0 as well as Uˆl = U j = M Ω for all ˆl D Lh (j). 28 / 36
Error Analysis Based on the Maximum Principle The Maximum Principle Proof of The Maximum Principle Similarly, we have U jk = M Ω, k = 1, 2,..., m and U i = M Ω. But this contradicts the assumption U i M D < M Ω. The same argument also leads to the conclusion that U i = U j for all i J, i.e. U is a constant on J, provided that the condition (3) and relation U j = max i J U i 0 hold. Remark 1: For the 5 point scheme of, the condition (2) of the maximum principle holds. If is replaced by ( + b 1 x + b 2 y + c) with c < 0, the conclusion still holds if x and y are approximated by central difference operators (2h x ) 1 0x and (2h y ) 1 0y respectively. Remark 2: For uniformly elliptic operators with c < 0, one can always construct consistent finite difference operators so that the condition (2) of the maximum principle holds for sufficiently small h, noticing that the second order difference operator has a factor of O(h 2 ), while the first order difference operator has a factor of O(h 1 ). 29 / 36
The Maximum Principle Apply the maximum principle to U, we have Corollary Suppose J and L h satisfy the conditions (1) and (2) in Theorem 1.2. Suppose that the grid function U satisfies L h U j 0, j J Ω. Then U can not take nonpositive minima on a interior node, i.e. { } m Ω min U i min min U i, 0. i J Ω i J D If L h satisfies further (3): J is connected with respect to the operator L h, and there exists an interior node j J Ω such that U j = min i J U i 0, Then, U must be a constant grid function on J.
Error Analysis Based on the Maximum Principle The Maximum Principle The Existence Theorem Theorem Suppose the grid J and the linear operator L h satisfy the conditions (1) and (2) of the maximum principle. Then, the difference equation { L h U j = f j, j J Ω, has a unique solution. U j = g j, j J D, 31 / 36
Error Analysis Based on the Maximum Principle The Maximum Principle Proof of the Existence Theorem We only need to show that L h U j = 0, j J U j = 0, j J. In fact, by the maximum principle L h U 0 implies U 0, and by the corollary of the maximum principle, L h U 0 implies U 0, thus U 0 on J. 32 / 36
Error Analysis Based on the Maximum Principle The Maximum Principle ( L h ) 1 is a Positive Operator Consider the discrete problem { L h U j = f j, j J Ω, Corollary U j = g j, j J D. Suppose the grid J and the linear operator L h satisfy the conditions (1) and (2) of the maximum principle. Then, and f j 0, j J Ω, g j 0, j J D, U j 0, j J; f j 0, j J Ω, g j 0, j J D, U j 0, j J; 33 / 36
Error Analysis Based on the Maximum Principle The Maximum Principle ( L h ) 1 is a Positive Operator The corollary says that ( L h ) 1 is a positive operator, i.e. ( L h ) 1 0. In other words, every element of the matrix ( L h ) 1 is nonnegative. In fact, the matrix L h is a M matrix, i.e. the diagonal elements of A are all positive, the off-diagonal elements are all nonpositive, and elements of A 1 are all nonnegative. 34 / 36
Error Analysis Based on the Maximum Principle The Comparison Theorem and the Stability The Comparison Theorem and the Stability Theorem Suppose the grid J and the linear operator L h satisfy the conditions (1) and (2) of the maximum principle. Let the grid function U be the solution to the linear difference equation { L h U j = f j, j J Ω, U j = g j, j J D. Let Φ be a nonnegative grid function defined on J satisfying L h Φ j 1, j J Ω. Then, we have max U j max U j + max Φ j max f j. j J Ω j J D j J D j J Ω 35 / 36
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