Stable finite difference schemes for the magnetic induction equation with Hall effect

Transcription

1 Eigenössische Technische Hochschule Zürich Ecole polytechnique féérale e Zurich Politecnico feerale i Zurigo Swiss Feeral Institute of Technology Zurich Stable finite ifference schemes for the magnetic inuction equation with Hall effect P. Corti an S. Mishra Research Report No. -3 April Seminar für Angewante Mathematik Eigenössische Technische Hochschule CH-89 Zürich Switzerlan

2 STABLE FINITE DIFFERENCE SCHEMES FOR THE MAGNETIC INDUCTION EQUATION WITH HALL EFFECT PAOLO CORTI AND SIDDHARTHA MISHRA Abstract. We consier a sub-moel of the Hall-MHD equations: the so-calle magnetic inuction equations with Hall effect. These equations are non-linear an inclue thir-orer spatial an mie erivatives. We show that the energy of the solutions is boune an esign finite ifference schemes that preserve the energy bouns for the continuous problem. We esign both ivergence preserving schemes an schemes with boune ivergence. The schemes are compare on a set of numerical eperiments that emonstrate the robustness of the propose schemes.. Introuction Plasmas are increasingly becoming important in a variety of fiels like astrophysics, solar physics, electrical an aerospace engineering. Specific problems inclue the stuy of supernovas, accretion isks, waves in the solar atmosphere, magnetic confinement fusion, the esign of plasma thrusters for spacecraft propulsion an of circuit breakers in the electrical power inustry. It is stanar to moel plasmas as magnetize fluis with flui motion shaping an in turn being shape by magnetic fiels. The base moel for such interaction are the equations of Magneto-Hyro Dynamics MHD:. ρ = ρu, t ρu. = t { ρu u + p + B {E + p B u + E B } I 3 3 B B, E.3 t =.4 B = E. t Here ρ, u, p are the gas ensity, velocity an pressure respectively. E an B are the electric an magnetic fiels. The total energy E is given by the equation of state:.5 E = p γ + ρ u + B. Here, γ is the gas constant. Equations from. to.3 represent the conservation of mass, momentum an energy an.4 escribes the Mawell s equations for the evolution of the magnetic fiel. In aition, the absence of magnetic monopoles implies that the MHD equations have to obey the ivergence constraint:.6 B =. We nee to escribe the electric fiel in.4 to complete the MHD equations. Different choices for the electric fiel Ohm s law lea to ifferent MHD moels. The most popular MHD moel is the ieal MHD equation where the electric fiel is given by.7 E = u B. }, Date: April 8,.

3 PAOLO CORTI AND SIDDHARTHA MISHRA The ieal MHD equations are etremely successful in several applications, see [] for an overview an [3] for a recent application in solar physics. However a major limitation of the ieal MHD equations is the requirement that the magnetic fiel lines are frozen into the flui. In many interesting applications in both astrophysics an engineering, we observe magnetic reconnection i.e. the magnetic topology changes uring the flow [7]. In orer to inuce reconnection, a possible mechanism is magnetic resistivity resulting in the Ohm s law:.8 E = u B + ηj. Here J is the current ensity an η is the resistivity parameter. The resulting equations are terme the resistive MHD equations. However, the resistive MHD equations o not suffice in moeling fast magnetic reconnection. A more effective alternative is to inclue the Hall effect [7, ]. The resulting Ohm s law is.9 E = u B + ηj + δ i J B + L ρ δe L [ J ρ. t ] +u J. Here L is the normalizing length unit, an δ e an δ i enote electron an ion inertia respectively; they are relate to electron-ion mass ratio by δe δ i = me m i. The term J B is the so-calle Hall term an J t +u J is the electron inertia term [7]. The equations nee to be complete by specifying Ampére s law for the current:. J = B. The MHD equations together with Ohm s law.9 an the above Ampére s law are terme as the Hall MHD equations. The Hall MHD equations are non-linear, high-orer equations an are etremely complicate to stuy in a mathematically rigorous manner. There have been various numerical stuies of the Hall MHD equations in [3, ] an references therein. However, all these papers tackle the problem from a computational point of view an o not inclue any rigorous results. In contrast to the above papers where schemes were esigne for the full Hall MHD equations, we aopt a ifferent approach. First, we consier a sub-moel: the Hall inuction equations,. t [ ] δe B + L ρ B = u B η B δe L ρ u B δ i B B. L ρ The Hall inuction equations are augmente with the ivergence constraint.6. Here, the unknowns are the magnetic fiel B an the velocity fiel u is specifie a priori. The parameters are as before. Observe that the Hall inuction equations are still non-linear with the non-linearity being present in the Hall term. Furthermore, the equations contain secon-orer spatial erivatives resistivity an thir-orer spatial an spatio-temporal erivatives electron inertia. In this paper, we investigate the Hall inuction equations from a mathematical point of view an erive stability estimates. The key tool will be to symmetrize the avection terms in. using the ivergence constraint. See [, 6, 5, 8] for the use of this technique for the ieal magnetic inuction equations an the resistive inuction equations. We then erive energy estimates in the Sobolev space H for the Hall inuction equations with smooth velocity fiels. We also erive stable finite ifference schemes for the Hall inuction equations. in this paper. These schemes preserve a iscrete version of the energy estimate. In particular, we iscuss both ivergence preserving schemes an schemes that are base on the symmetric form of the equation an may not preserve a iscrete version of the ivergence constraint. Numerical eperiments emonstrating the robust performance of the propose schemes are presente. To the best of our knowlege, the propose schemes are the first set of rigorously proven stable schemes for the Hall inuction equations. The rest of the paper is organize as follows: in section, we stuy the continuous problem for the Hall inuction equations an show energy estimates. Stable numerical schemes that satisfy a iscrete version of the energy inequality are presente in section 3 an we present numerical eperiments in section 4.

4 FD FOR HALL MAGNETIC INDUCTION 3. The Continuous Problem The main aim of this section is to erive energy estimates for the continuous problem corresponing to the Hall inuction equations.. It turns out that the avection terms in. are not symmetric an impair the erivation of an energy estimate. We will symmetrize this term see [, 6, 5] by using a the vector ientity. u B = B u B u+u B u B. We use the ivergence constraint.6 an subtract u B from. to obtain the symmetric form of the Hall inuction equations: [ ] δe B + t L ρ B =B u B u u B η B. δe ρ u B δ i B B. L ρ L We show the following energy estimate for the symmetric form of the Hall inuction equations, Theorem.. Let the velocity fiel u C R 3. Furthermore, assume that the solution B of. ecays to zero at infinity, then following apriori estimate hols: B L t R 3 + δe L ρ B L R 3 C B L R 3 + δe.3 L ρ B L R 3 with C being a constant that epen on u an its erivatives only. Proof. To simplify the notation an emphasize the symmetry of the equation, we rewrite the term B u in the matri form obtaining: B t + δe L ρ B =CB ub u B η B t δe ρ u B δ i B B L ρ where.4 L C = u y u z u u y u z u u 3 y u 3 z u 3 To obtain the L estimate, we multiply the equation with B an then we integrate over R 3 resulting in B δe + R t L 3 ρ B B t = [B CB ub R u B ηb B 3 δe ρ B u B δ i B B B]. L ρ Partial integration yiels t B L R 3 + δe L L ρ B L R 3 = R 3 [B CB ub η B + δe L. ρ u B δ i B B B ] L ρ }{{} =

5 4 PAOLO CORTI AND SIDDHARTHA MISHRA = [B CB R ub η B + δe L 3 ρ u B ]. We i not consier the bounary terms because they vanish since the solution B is ecaying to zero at infinity. Using Cauchy-Schwartz we obtain B L t R 3 + δe B L L R 3 C A B L R + C δe 3 B L ρ B L R 3 u where C A ma i L k,i k an C B = u R 3 L R 3. We have shown that the inuction equation in symmetric form possesses an energy estimate. However, the ivergence of the solution of. is not preserve eactly. Nevertheless, we have the following estimate: Theorem.. Let u C R 3 Furthermore, assume that the ivergence of the solution of., B, ecays to zero at infinity, then the following apriori estimate hols:.5 t B L R 3 C B L R 3 with C being a constant that epen on u an its erivatives only. Proof. Applying the ivergence operator on. we obtain B.6 = u B t using vector ientity we can rewrite the right han sie of the equation B = u B B u. t Multiplying this equation by B an integrating it over R 3 we obtain t B L R 3 = u B u B, R 3 R 3 with partial integration we obtain t B L R 3 = u B u n B s R 3 R 3 u L R 3 B L R 3. Corollary.. If the conitions for theorems theorem.. an theorem.. hol an the initial ata B H R 3, then the estimates imply that B L,T,H R 3. Remark.. The ivergence transport equation.6 implies that the ivergence remains zero if the initial ata has zero ivergence. In this case, the solutions of the symmetric form. are also weak solutions of the non-symmetric form. of the Hall inuction equations. 3. Numerical Scheme In this section, we esign numerical schemes that satisfy a iscrete version of the energy estimates of the last section. The computational omain is Ω = [,L ] [,L y ] [,L z ] an we efine a uniform mesh of N times N y times N z points with coorinates i = i, y j = j y an z k = k z. In this case = L /N, y = L y /N y an z = L z /N z are mesh withs. The point values of the magnetic an velocity fiels are B i,j,k B i,y j,z k ũ i,j,k u i,y j,z k.

6 FD FOR HALL MAGNETIC INDUCTION 5 We will provie a very general iscrete formulation an specify the necessary requirement that the iscrete erivative shoul posses. As in [6, 5, 8], the key requirement for the iscrete erivative is to satisfy a summation by parts SBP conition. The eact form of operators satisfying the requirements are presente in appeni A. We start with one imensional iscrete erivatives using gri functions in vector form, i.e., w =w, w N. An approimation of the spatial erivative, D possesses the summation by parts property see [] if it can be written as D = P Q, where P is a iagonal N N positive efinite matri an Q an N N matri satisfying: 3. Q + Q = R N L N where R N an L N are N N matrices: iag,,, an iag,,,. The operator P efines an inner prouct 3. v, w P = v P w with the associate norm w P =w, w / P that is equivalent to the norm w = k w k /. Net, we efine averaging operators such that we can obtain an approimate form of the chain rule. We efine symmetric averaging operators as 3.3 A w i = α k w i+k with q α k = an α i = α i. In [8], the following iscrete chain rule was shown, Lemma 3.. Given any smooth function u, we enote his restriction on the gri as ū an let be w a gri function. Then we can efine an average operator Ā couple to D such that 3.4 D ū w = ū D w+d ū Āw + w, where u w i,j,k = u i,j,k w i,j,k an with Āw = q kβ kw i+k an w P C w P. Proof. The iscrete ifferential operator acting on a gri function w i can also be written through sums: D w i = q β k w i+k. with q β k = an q kβ k =. Then the resiual w is given by w i = D ū w i ū D w i D ū Āw i = β k ū i+k w i+k ūi β k w i+k β l ū i+l We epan ū with Taylor-epansion l=r l=r ū i+k =ū i + kū i + k c i k where c i k =ū i ξ k obtaining w i = ūi β k w i+k +ū ūi i kβ k w i+k + k β k c i kw i+k ūi ūi ū i + l β l c i l kβ k w i+k = γkw i i+k. Here γk i = kβ kc k q l=r l β l c i l k. Since the γk i are boune, we have that w P C w P kβ k w i+k. β k w i+k

7 6 PAOLO CORTI AND SIDDHARTHA MISHRA where C epens only on the maimum of γ i k an on the norm P. We show the following lemma claiming that ifferential operators commute with average operators, Lemma 3.. The iscrete ifferential operator D an the average operator A commute. Proof. We write the iscrete ifferential operator acting on a gri function w i through sums: D w i = Then applying the two operators consecutively we obtain D A w i = = = q q k =r β k A w i+k = q k =r α k q k =r β k.w i+k. q k =r β k β k α k w i+k +k = k =r β k w i+k +k = q α k w i+k +k k =r β k α k w i+k +k α k D w i+k =A D w i We use the above one imensional operators to buil the multiimensional operators i.e, mappings of three imensional gri functions w i,y j,z k =w i,j.k to a column vector 3.5 w =w,,,w,,,,w,,nz,w,,,,w N,N y,n z. We efine the iscrete ifferential operators an averages = D A y A z, y = A D y A z, z = A A y D z, A = Ā A y A z, A y = A Āy A z, A z = A A y Āz. Here is the Kronecker prouct. We also eten the inner prouct by P = P P y P z with the corresponing norm w P =w, w / P. We generalize the one imensional operators to three imensions using the average operators A, A y an A z. Setting the averaging operators to the ientity mapping allows us to recover the stanar version of one imensional iscrete operators. However, we will use a more general form of the averaging operator that will allow us to inclue a larger group of ifference operators incluing the ivergence preserving operators of [] an some of the ivergence preserving operators propose in [9]. We can epan lemma 3. in three imensions: Corollary 3.. Given any smooth function ū as above, then 3.6 u w u w P C w P where C a constant which epens on the first orer erivative of u. Proof. The proof of this corollary is similar to the one one before only using a substitution with a Taylor epansion of egree one. The summation by parts property of the ifferential operators, couple with the inner prouct P will result in a iscrete version on integration by parts:

8 FD FOR HALL MAGNETIC INDUCTION 7 Lemma 3.3. For any gir function v an w, we have 3.7 v, w P + v, w P = v R L w, v, y w P + y v, w P = v U Dw, v, z w P + z v, w P = v T Bw where R = R N P y A y P z A z, L = L N P y A y P z A z, U = P A R Ny P z A z, D = P A y L Ry P z A z, T = P A y P y A y R Nz an B = P A y P y A y L Nz. Proof. Since A k s are symmetric an P k s iagonal, we can calculate v, w P + v, w P = v P P y P z P Q A y A z w + P Q A y A z v P P y P z w = v Q A y P y A z P z w + v Q P y A y P z A z w = v Q + Q P y A y P z A z w = v R N P y A y P z A z w v L N P y A y P z A z w. The proof for the other space irections follow analogously. The right han sie of the above equation represents the evaluation of the gri function on the bounary of Ω. Until now all our analysis was for scalar operators, we eten it to vector-value iscrete ifferential operators below. Corollary 3.. We erive from the scalar summation by parts rules for scalar fiels ṽ, D w P = Dṽ, w P + i ˆv i S i ŵ i, 3.8 ṽ, D w P =D ṽ, w P + i,j,k ɛ i,j,kˆv i S j ŵ k where D =, y, z, S = L R, S = U D, S 3 = T B an ɛ i,j,k is the Levi-Civita symbol. Proof. The proof of this corollary is given by the irect application of the previous theorem on iscrete vector fiels. In the continuous case, the key property for the preservation of ivergence is the ientity: w = for all w C R 3 3. From Lemma 3. an the efinition of Kronecker prouct, one observes that the ifference operators, y an z commute, we can show that is also the case for the iscrete ifferential operator: Corollary 3.3. Every gri function ŵ i,j,k couple with D =, y, z satisfies 3.9 D D ŵ =. The proof of this corollary is straightforwar. Generalizing lemma 3.3 presente in [6] for the vector operator D: Lemma 3.4. If ũ is a vector gri function 3. v, ũ D ṽ P = ṽ, ũd ṽ i Dũ i ṽ Proof. See the proof in [5] an apply it on each component of ũ D. P + ṽ S i ũ i ṽ i

9 8 PAOLO CORTI AND SIDDHARTHA MISHRA Now we have all the ingreients to present two ifferent classes of numerical schemes. The first set of schemes terme as symmetric schemes will iscretize the symmetric version of the Hall magnetic inuction equation.. The secon set of schemes will iscrete the non-symmetric version. of the Hall inuction equations an will preserve a iscrete version of ivergence. Hence, we term it ivergence preserving scheme. 3.. Symmetric Scheme. We iscretize the symmetric form. of the Hall inuction equations an efinie a semi-iscrete numerical scheme as δe B i,j,k + t L ρ D D B i,j,k = AV B, ũ ũ i,j,k D B i,j,k ηd D B i,j,k δe L ρ D ũ i,j,k DD B i,j,k δ i 3. L ρ D D B i,j,k B i,j,k where AV B, ũ =Ā B i,j,k Dũ i,j,k Ā B i,j,k ũ i,j,k Āy B i,j,k y ũ i,j,k Āz B i,j,k z ũ 3 i,j,k, Ā B i,j,k =Ā B i,j,k, Āy B i,j,k, Āz B 3 i,j,k. The term AV represent the iscretisation of B u B u. We estimate it below, Lemma 3.5. For B i,j,k gir function an ũ i,j,k a boune gri function, we have 3.4 B, AV B, ũ P C B P where C epens on ũ an its iscrete erivative only. Proof. We write 3.4 component-wise an using the triangle inequality, we obtain 3 B, AV B, ũ P B i, Ā B ũ i P + B i, Āy B y ũ i P + B i, Āz B 3 z ũ i P i= + B i, Ā B i ũ P + B i, Āy B i y ũ P + B i, Āz B i z ũ 3 P. Since the iscrete erivative of ũ are boune, we can etract its maimum from the P norms an obtain 3 B, AV B, ũ P C B i, Ā B P + B i, Āy B P + B i, Āz B 3 P + B i, Ā B i P i= + B i, Āy B i P + B i, Āz B i P C B, Ā B P + B, Āy B P + B 3, Āz B 3 P + B, Ā B P + B 3, Ā B P + B, Āy B P + B 3, Āy B P + B, Āz B 3 P + B, Āz B 3 P Take the first term an first use the equivalence of inner prouct, then write the average operator through summation, an finally using the Cauchy inequality yiels B, Ā B P C B, Ā B = C B i,j,kα l B i,j+l,k C ma lα l i,j,k,l [ B i,j,k + B i,j+l,k ] C B. i,j,k,l Repeating this proceure for all the term of the sum, yiels to B, AV B, ũ P C B. The use of the equivalence between Eucliean an P norms conclue the proof of lemma. This lemma will allow us to is now prove the energy stability of the scheme. The main theorem is:

10 FD FOR HALL MAGNETIC INDUCTION 9 Theorem 3.. Let ũ i,j,k = u i,y j,z k be the point evaluation of a function u C an let the approimate solutions B of 3. go to zero at infinity, then the following estimates hol B δe P + t ρ D B P C B δe 3.5 P + ρ D B P with C a constant that epen on u an its erivative only. L Proof. Multiplying both sies of the scheme 3. by B yiels t B δe P + L ρ B, D D t B P = B, AV B, ũ P B, ũ D B P η B, D D B P δe L ρ B, D ũ D D B P δ i L δ B, D D B B P. Using summation by parts of corollary 3. an lemma 3.4 B δe P + t L ρ D B P = B, AV B, ũ P B, ũ D B Dũ i B P i η D B δe P L ρ D B, ũ D D B P δ i L δ D B, D B B P = }{{} = L B, AV B, ũ P B, ũ D B i Dũ i B P η D B P δe L ρ D B, ũ D D B i Dũ i D B P. All the bounary terms have been neglecte since the ata ecay to at infinity. Using lemma 3.4 an lemma 3.5 we obtain B δe P + t ρ D B P C B δe P + ρ D B P. L L Since the ivergence is not preserve by the symmetric scheme, we show that the ivergence generate by the symmetric scheme is boune a similar result for the ieal magnetic inuction equations was shown in [8]. Theorem 3.. Let ũ i,j,k = u i,y j,z k be the point evaluation of a function u C an let the solutions of 3. go to zero at infinity, then the following estimates hol 3.6 t D B P C D B P + B with C a constant that epen on u an an its erivative an on the regularity of the gri. Proof. We efine the iscrete ivergence ˆω = D B an using the numerical scheme 3. with corollary 3.3, we obtain an equation for its evolution t ˆω = D AV B, ũ ũ D B. Now epaning each component of AV an using lemma 3., we obtain, for eample for the first component: AV B, ũ = Āy B y ũ + Āz B 3 z ũ Āy B y ũ Āz B z ũ 3 = y B û B û ū ω +ũ D B + R y B, B +R z B, B 3

11 PAOLO CORTI AND SIDDHARTHA MISHRA where the resiual terms are R y B, B i,j,k = y R z B, B 3 i,j,k = z l=r l=r γ j l ũ B i,j+l,k γ j l ũ B i,j+l,k γ j l ũ 3 B i,j,k+l γ j l ũ B 3 i,j,k+l. Here the γk i are efine in the proof of lemma 3. an epen on the velocities u. Applying the same technique on the two other components we obtain where AV B, ũ ũ D B =D ũ B ũ ˆω + R R = R y B, B +R z B, B 3 R B, B +R z B, B 3 R B 3, B +R y B 3, B Here R P C ma, y, z B P. Setting these results in the iscrete evolution equation an using corollary 3.3 we obtain ˆω = D ˆω ũ +DR. t The time evolution of the P norm of the ivergence is t ω P = ω, t ω P = ω, D ω ũ P + ω, DR P, with summation by parts an lemma 3.4 we obtain t ω P = D ω, ω ũ P + ω, DR P = ũd ω, ω P + ω, DR P =ũd ω, ω P D ũ ω, ω P + ω, DR P C ω P + ω P DR P 3.7 C ω P + C min, y, z ω P R P C ω P + C ma, y, z min, y, z ω P B P where the C s epen on u an its erivative. To obtain this result the bounary terms are neglecte since B ecays to at infinity. We have also use that ˆ, ṽ ŷ P =ṽ ˆ, ŷ P where ˆ an ŷ are scalar gri function an ṽ a vector gir function. This is the case since P is a iagonal matri. We conclue the proof using Cauchy s inequality. Although the symmetric scheme 3. is stable in H, it might generate small but boune ivergence errors. Net, we esign a scheme that preserves a iscrete version of the ivergence operator. 3.. Divergence Preserving Schemes. The symmetric scheme oes not preserve a iscrete version of the ivergence constraint as it iscretizes the symmetric version. of the Hall inuction equations. We have to iscretize the non-symmetric stanar version. of the Hall inuction equations in orer to esign a ivergence preserving scheme. Such a scheme for is δe B i,j,k + t L ρ D D B i,j,k = D ũ B i,j,k ηd D B i,j,k δe L δ D ũ i,j,k DD B i,j,k δ i 3.8 L δ D D B i,j,k B i,j,k. The application of corollary 3.3 shows that the above scheme clearly satisfies the ivergence constraint: 3.9 t D B =..

12 FD FOR HALL MAGNETIC INDUCTION The proof for the energy stability of this scheme is more comple since the equation is not symmetric. We introuce the following one-sie operators: Dw s i = w i+s w i 3.. s The operators Dy s an Dz s are efine analogously. Lemma 3.6. For two gri functions u an w, the following ientity hols, 3. D u w =u D w + ÂD s u,w, with ÂD s u,w i = k kβ kd k u i w i+k is an average over iscrete one sie erivative of u multiplie with w. Proof. We compute the ifference D u w i u D w = = noting that it takes the form of the esire average. k k β k u i+k w i+k u i β k k u i+k u i w i+k, k The energy boun for the ivergence preserving scheme is given below: β k w i+k Theorem 3.3. Let ũ i,j,k = u i,y j,z k be the point evaluation of a function u C an let the solutions of 3.8 with an initial ata with D B =. If the approimate solution ecays to zero at infinity, then the following estimates hol 3. t B P + δe with C a constant that epens on u an its erivative only. L ρ D B P C B δe P + L ρ D B P Proof. To prove the energy estimate, we have to symmetrize the avection terms in the scheme. This is possible since the metho preserve ivergence; in this case we can subtract form 3.8 ṽ D B =. Using lemma 3.6, we cam reformulate the iscrete avection part with R B, ũ i,j,k = D ũ B ṽ D B =ũ D B + R B, ũ, A y [ s yû ˆB i,j,k s yû ˆB i,j,k ]+A z [ s zû ˆB 3 i,j,k s zû 3 ˆB i,j,k ] A [ s yû ˆB i,j,k s û ˆB i,j,k ]+A z [ s zû ˆB 3 i,j,k s zû 3 ˆB i,j,k ] A [ s û 3 ˆB i,j,k s û ˆB 3 i,j,k ]+A y [ s yû 3 ˆB i,j,k s yû ˆB 3 i,j,k ] The resulting scheme is δe B i,j,k + t L ρ D D B i,j,k = ũ D B + Rũ, B ηd D B i,j,k δe L δ D ũ i,j,k DD B i,j,k δ i L δ D D B i,j,k B i,j,k We see that this is very similar to result obtaine for the symmetric case. The only ifference is that, instea the average term B, AV B, u P we have a resiual term B, R B, u P. Then showing that the resiual term is boune B, R B, u P C B P will conclue the proof. We can follow the same proceure we use in the proof of lemma 3.5 to boun the avection term. k.

13 PAOLO CORTI AND SIDDHARTHA MISHRA The proof follows the theory presente in [9] where a generalize finite volume scheme that preserve iscrete ivergence is presente. The metho presente there is more general in its formulation, an inclues a large class of alreay known ivergence preserving methos Time Stepping. Both the semi-iscrete symmetric scheme an ivergence preserving schemes can be written as 3.3 ˆB + αd D ˆB = RHSˆB, û, t δ with α = e L ρ an the function RHS will epen on the scheme. We use a stanar Runge Kutta metho to upate the solution in time. Even though we use eplicit RK methos, we have to solve linear equations corresponing to the lhs of the above scheme at each time step. As an eample, we consier a secon orer SSP metho [4]: 3.4 ˆB = ˆB n + ta RHSˆB n, û, ˆB = ˆB + ta RHSˆB, û, ˆB n+ = ˆB n + ˆB. Here, A = I + αf, F is the matri representation of D D an I is the ientity matri. In this paper, we are using a irect solver. However, the matrices F an A are of size 3 N N y N z, an not well conitione for high-resolution meshes. The esign of an efficient preconitioner for the matri A is a subject of ongoing research. 4. Numerical Eperiments For simplicity, we consier the Hall inuction equations in two space imensions an present numerical eperiments comparing ifferent schemes propose above. In two space imensions, the symmetric version. reas as 4.a 4.b t [ ] δe ˆB + ˆB = L ρ Ĉ ˆB û ˆB η ˆB δe L ρ û ˆB [ δe B 3 t L ρ B 3 Here, ˆB =B,B an û =u,u. imensions: 4.a 4.b ] δ i L ρ ˆB B 3, = C B û B 3 + η B 3 + δ i L ρ ˆB ˆB. δe L ρ û B 3 We have also introuce a compact curl operator in two v := v v v y ψ ψ := y ψ, where ˆv : R 3 R an ψ : R 3 R. We consier the following numerical eperiments:

14 FD FOR HALL MAGNETIC INDUCTION Pure avection. First, we test the propose numerical schemes for the magnetic inuction equations without Hall, electron inertia an resistivity terms. We take the velocity fiel u =y, in the following. Then, 4. with η = δ i = δ e = has an eact solution see [] given by 4.3 ˆB, y, y =RtˆB Rt.y, with Rt a rotation matri with angle t. The initial ata is 4.4 ˆB, y =4 y e / +y in the computational omain Ω =[.5,.5] [.5,.5]. We consier Neumann type non-reflecting bounary conitions. The eact solution represents the rotation of the initial hump aroun the omain with the hump completing one rotation in the perio T =π. We will test the following four schemes: the secon- an fourth-orer versions of the symmetric scheme 3. with ifference operators given in appeni A, secon- an fourth-orer version of the ivergence preserving scheme 3.8. The convergence plots in L are shown in figure. Symmetric Scheme Cent. Diff. n or. Cent. Diff. 4n or. Divergence Preserving Scheme Cent. Diff. n or. Cent. Diff. 4n or.!! Error in L norm! Error in L norm!!3!3!4 3 a Symmetric schemes!4 3 b Divergence preserving schemes Figure. Convergence plots for the avection problem for the ifferent schemes analyze. For time integration we have use a secon orer SSP an a stanar thir orer Runge-Kutta metho. The results are obtaine using ifferent mesh sizes, form 8 to points. The eperimental convergence orers are shown in Table an emonstrate that the epecte orers of accuracy are obtaine in practice. n or 4 or Preserving Symmetric Table. Convergence rates for the avection Problem.

15 4 PAOLO CORTI AND SIDDHARTHA MISHRA 4.. Force Solutions. In orer to test the convergence rates for various schemes for full Hall inuction equations, we a a forcing term such that the rotating hump 4.3 remains a solution of the force equations. The Hall inuction equations with the forcing term are t [ δe ˆB + ˆB L ρ We forcing term Ŝ is [ δe B 3 t L ρ B 3 ] ] = Ĉ ˆB û ˆB η ˆB δe L ρ û ˆB δ i L ρ ˆB B 3 +Ŝ, y, t, = C B û B 3 + η B 3 + δe L δ i L ρ ˆB ˆB + S 3, y, t. ρ û B 3 Ŝ, y, t = 6 P, y, t η e cost+y sint/ +y cost sint sint y cost S 3, y, t =. Here P, y, t = cost + y sint y 3. The convergence results for four ifferent schemes are presente in figure. The obtaine orers of convergence are shown in Table. Again, the epecte orers of convergence are obtaine.,! Symmetric Scheme Cent. Diff. n or. Cent. Diff. 4n or.! Divergence Preserving Scheme Cent. Diff. n or. Cent. Diff. 4n or.!! Error in L norm Error in L norm!3!3!4 3 a Symmetric schemes!4 3 b Divergence preserving schemes Figure. Convergence plots for the Force Problem with η =.,δ i =. an δ e =4.5 n or 4 or Preserving Symmetric Table. Convergence rates for the Force Problem with η =.,δ i =. an δ e =4.5

16 FD FOR HALL MAGNETIC INDUCTION Unforce solutions. In the final numerical eperiment, we test the full Hall inuction equations without any forcing for the rotating hump problem. We set η =., δi =. an δe = 4.5 an compute the solutions on a mesh 6 6 points. We compare the results with those obtaine for the pure avection of the hump. The results are shown in figure 3 an emonstrate the robustness of the secon-orer symmetric scheme. Similar results were obtaine with the ivergence preserving scheme. The results show that the aition of resistivity, electron inertia an Hall effect leas to iffusion of the original hump an the creation of a non-zero B3 component even if the initial B3 is set to zero. B.5 B !.5.5 y y...5!.5!.!.!!!.!.5!.5!.4!.3!!.5!.5!!.4!!.5!!.5.5.5!.5!.5.5 a B component for the avection problem. B.5!.6!!.5!! b B component for the force problem B !.5.5 y y...5!.5!.!.!!!.!.5!.5!.3!!.4!!.6!.5!.5!!.5!! !.5!.5!.4 c B component for the avection problem. B !.!!.4!!.5!.6!.5!!.8!!.5!! ! e B3 component for the avection problem.! !4 B3!.5!.5!.5!.5.5!.5 B component for the force problem. y y.5!!.5!!.5!.5!!.5!! ! f B3 component for the force problem. Figure 3. On the right plots for solution with η =.,δi =. an δe = 4.5 after T = π an on the left the avecte solution after the same time.

17 6 PAOLO CORTI AND SIDDHARTHA MISHRA We conclue by tabulating the iscrete ivergence generate by the schemes for this problem in Table 3. As epecte, the ivergence preserving scheme preserves ivergence to machine precision. On the other han, the symmetric scheme oes generate some spurious ivergence. Symmetric Preserving 5 4.6e 6.4e7 5.4e 5 8.6e e 7 9.e 7.5e 7.e 6 5.4e 7.e 6 Table 3. Discrete ivergence for the unforce problem. The ivergence errors converge quite rapily to zero as the mesh is refine. Furthermore, there was no noticeable ifference in the quality of the results for the primary solution variables between the symmetric an ivergence preserving schemes. 5. Conclusion We consier the Hall inuction equation, a sub-moel for the Hall MHD equations, in this paper. An energy estimate is erive for the equations an numerical schemes that satisfy a iscrete version of this estimate are presente. A special class of numerical schemes also satisfy the ivergence constraint. The propose schemes are teste on a set of numerical eamples an are emonstrate to be robust. The etension of the schemes to the full Hall MHD equations is the subject of forthcoming papers. Appeni A. Finite Difference Operators The ifferent operators use in our numerical eperiment, are base on one imensional operators couple together with Kronecker prouct. The one imensional operators are given for q =, y, z in matri form: Secon orer central ifference D q = Pq Q = , P q = q q Fourth orer central ifference D q 4 = Pq Q = , P q q = q Combining this operators we obtain the two spatial iscretisation use in the numerical eperiments. We give the iscrete erivative for the irection, the ones for the other spatial irections are efine analogously. Stanar secon an fourth orer operator are where I q are the ientity matrices. = D k I y I z k =, 4.

18 FD FOR HALL MAGNETIC INDUCTION 7 References [] D. Biskamp, Nonlinear Magnetohyroynamics Cambrige University Press, New York, 993. [] F.G. Fuchs, K. Karlsen, S. Mishra an N.H. Risebro. Stable upwin schemes for the magnetic inuction equation. Mathematical Moeling an Numerical Analysis,435, 85-85, 9. [3] F.G. Fuchs, A.D. McMurry, S. Mishra, N.H. Risebro an K.Waagan Simulating waves in the upper solar atmosphere with SURYA: a well-balance high-orer finite volume coe. Astrophysical Jl., To appear. [4] S. Gottlieb, C-W.Shu an E. Tamor, Strong stability-preserving high-orer time iscretisation metho, SIAM Review, 43, 89-,. [5] U. Koley, S. Mishra, N.H. Risebro an M. Svär, Higher orer finite ifference schemes for the magnetic inuction equations with resistivity. IMA Jl. Num. Anal.,, To appear. [6] U. Koley, S. Mishra, N.H. Risebro an M. Svär, Higher orer finite ifference schemes for the magnetic inuction equations. BIT Numerical Mathematics, 49,, , 9. [7] Z.W. Ma an A. Bhattacharjee, Hall magnetohyroynamic reconnection: The Geospace Environment challenge. Journal of Geophysical Research, 6, ,. [8] S. Mishra an M. Svär, On stability of numerical scheme via frozen coefficients an magnetic inuction equations, BIT Numerical Mathematics, 5,, 85-8,. [9] S.Mishra an E.Tamor, Constraint preserving schemes using potential-base flues. I. Multiimensional transport equations, Communication in Computational Physics,,9,3,688-7,. [] X. Qian, J. Balbás, A. Bhattacharjee an H. Yang A Numerical Stuy of Magentic Reconnection: A Central Scheme for Hall, Hyperbolic Partial Differential Equations, Theory, Numerics an Applications, Proceeings of the th. international conference hel at the University of Marylan. [] M. Svr an J. Norstrm On the orer of accuracy for ifference approimations of initial-bounary value problems, Journal of Computational Physics vol,8, ,6. [] M. Torrilhon an M. Fey Constraint-preserving upwin methos for multiimensional avection equations. SIAM Journal of Numerical Analysis,4,4,694-78,4. [3] G. Toth, Y. J. Ma an T. I. Gombosi Hall Magnetohyroynamics on Block Aaptive Gris. Journal of Computational Physics,7, ,8. Paolo Corti Seminar For Applie Mathematics SAM ETH Zentrum Rämistrasse 89 Zürich aress: paolo.corti@sam.math.ethz.ch Sihartha Mishra Seminar For Applie Mathematics SAM ETH Zentrum Rämistrasse 89 Zürich aress: sihartha.mishra@sam.math.ethz.ch

19 Research Reports No. Authors/Title -3 P. Corti an S. Mishra Stable finite ifference schemes for the magnetic inuction equation with Hall effect - H. Kumar an S. Mishra Entropy stable numerical schemes for two-flui MHD equations - H. Heumann, R. Hiptmair, K. Li an J. Xu Semi-Lagrangian methos for avection of ifferential forms - A. Moiola Plane wave approimation in linear elasticity -9 C.J. Gittelson Uniformly convergent aaptive methos for parametric operator equations -8 E. Kokiopoulou, D. Kressner, M. Zervos an N. Paragios Optimal similarity registration of volumetric images -7 D. Marazzina, O. Reichmann an Ch. Schwab hp-dgfem for Kolmogorov-Fokker-Planck equations of multivariate Lévy processes -6 Ch. Schwab an A.M. Stuart Sparse eterministic approimation of Bayesian inverse problems -5 A. Barth an A. Lang Almost sure convergence of a Galerkin Milstein approimation for stochastic partial ifferential equations -4 X. Claeys A single trace integral formulation of the secon kin for acoustic scattering -3 W.-J. Beyn, C. Effenberger an D. Kressner Continuation of eigenvalues an invariant pairs for parameterize nonlinear eigenvalue problems - C.J. Gittelson Aaptive stochastic Galerkin methos: beyon the elliptic case - C.J. Gittelson An aaptive stochastic Galerkin metho